APPENDIX

PSEUDORANDOM SIMULATIONS OF GLOBAL CIRCULAR LAW

In this appendix, we reproduce pseudorandom simulations of spectral properties of the product of

random matrices in the case where the dimension of the matrices is large and one matrix satisfies

the conditions of Circular Law. The domain of distribution of eigenvalues of these products

resembles a circle. Such matrices are used in many applied sciences, especially in the theory of

gyroscopes, control theory and some areas of physics.

Figure 1 shows the shape of the Sombrero probability density (see formula (26.42)):

( ) - 1 {b (a_b)2(t 2 +s 2 )} 2 2 (b)j b p t, s - 21Tab a + - Ja2b2+(a-b)2(t2+s 2)2 ,t + s < a + 2, a > 0, > O.

Observe that, for a = b, we have a cylinder. This figure was obtained by using Mathematica 4.0.

0.2

Figure 1: Sombrero probability density for a = 1, b = 7

464 Appendix

Figure 2 shows the shape of the Sombrero probability density (see formula (26.39)):

(X) [ 82 82 ] P (t, s) = _(47f)-1 Jo 8t2 + 8s2 f (y, t, s) dy,

where f (y, t, s) satisfies the canonical equation K 26

-1 3 { [It[2 + [S[2]C;2 }-1 f(y,t,s)=3 ~ y[l+f(y,t,s)]+ l+f(y,t,s) ,

y > 0, cp > 0, p = 1,2,3 and the circular domain of the Sombrero probability 3

density is equal to [(t, s) : t 2 + S2 ::::; 3- 1 L c~]. This figure was obtained by using p=l

Mathematica 4.0 and is available upon request.

Figure 2. Sombrero probability density for C1 = 1, C2 = 1.5, C3 = 2. 0.3 million pseudorandom realizations of Gaussian real matrices 3 30 = {~ij n~=l each multiplied by a diagonal matrix A30 of three diagonal blocks with diagonal entries C1 = 1, C2 = 1.5, C3 = 2 respectively were used to obtain this histogram.

Pseudorandom simulations 465

Figure 3 exhibits a simulation of eigenvalues from 1000 replications of Gaus­sian complex matrices 230 = {~ij }f,~=l each multiplied by a diagonal matrix A30

with diagonal entries in the upper half equal to 3 and the lower half equal to 5.

50 :".

30

10

-10

-30

-50 L-__ ~ __ ~~~ ____ ~ __ ~~~ __ ~ ____ ~ __ ~ __ ~

-50 -30 -10 10 30 50 Re(Z)

Figure 3. 1000 replications of eigenvalues of complex matrices 230A30

466 Appendix

Figure 4 exhibits a simulation of eigenvalues from 1000 replications of Gaus­sian real matrices 3 30 = {~ij }t,~=l each multiplied by a diagonal matrix A30 with diagonal entries in the upper half equal to 3 and the lower half equal to 5.

50

30

10

-10

-30

-50 -50 -30 -10 10 30 50

Re(Z)

Figure 4. 1000 replications of eigenvalues of matrices 330A30

REFERENCES

Although most of these references were not consulted in the writing of this book, they are listed here to give the reader an idea of the scope of the theory of random matrices and its applications.

Aizenman, M., Lebowitz, J. L. and Ruelle, D. [ALR] Some Rigorous Results on the Sherington-Kirkpatrick Spin Glass

Model, Commun. Math. Phys. 112, 3-20 (1987).

Anderson, T. W. [And1] Asymptotic theory for principal component analysis, Ann. Math.

Statist. 34, 122-148 (1963). [And2] An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley

and Sons, New York (1984).

Arharov, L. V. [Arh] Limit theorems for the characteristic roots of a sample covariance

matrix, Soviet Math. Dokl, 12, 1206-1209 (1971).

Arnold, L. [AmI] On the asymptotic distribution of the eigenvalues of random ma­

trices, 1. Math. Anal. Appl. 20, 262-268 (1960). [Am2) On Wigner's semicircle law for the eigenvalues of random matrices,

Z. Wahr. Verw. Geb. 34, 191-198 (1971).

Bahr, Von and Esseen, C.G [BE) Inequalities for the r-th absolute moment of a sum of random vari­

ables, 1 ::; r ::; 2, Ann. Math. Statist. 36, 1, 299-303 (1965).

Bai, Z. D. [Bail] Convergence rate of expected spectral distributions of large random

matrices. 1. Wigner Matrices, Ann. Probab. 21,2,615-648 (1993). [Bai2] Convergence rate of expected spectral distributions of large random

matrices. II. Sample Covariance matrices, Ann. Probab. 21, 2, 649-672 (1993).

Bai, Z. D. and Yin, Y. Q. [BY] Limiting behavior of the norm products of random matrices and

two problems of German-Hwang, Probab. Theory, Related Fields, 73, 555-569 (1986).

468 References

Bai, Z. D., Silverstein, J. W. and Yin, Y. Q. [BSY] A note on the largest eigenvalue of a large-dimensional sample co­

variance matrix, J. Multivar. Anal. 26,166-168 (1998).

Bai, Z. D. and Wu Y. [BWu1] Limiting behavior of M -estimators of regression coefficients in high

dimensional linear models. I. Scale-dependent Case, Journal of Multivariate Analysis 51,211-239 (1994).

[BWu2] Limiting behavior of M-estimators ofregression coefficients in high dimensional linear models. II. Scale-invariant Case, Journal of Mul­tivariate Analysis 51, 240-251 (1994).

Barsov, D. A. [Bar] Limit theorems for traces of sample covariance matrices, Theory

Probab. Appl .. 29,2,383-391 (1984).

Basor, E. L. [Bas] Distribution functions for random variables for ensembles of posi­

tive hermitian matrices, Communications in Mathematical Physics 188, 327-350 (1997).

Bateman, H. and Erdelyi, A. [Bat] Higher Transcendental Functions, vols. 1-3. McGraw Hill, New

York, (1953--1955).

Bellman, R. [Bel] Introduction to Matrix Analysis, McGraw Hill, New York, (1960).

Bennet, G., Goodman, V. and Newman, C. M. [BGN] Norm of random matrices, Pacific J. Math. 59, 359-365 (1975).

Berezin, F. A. [Ber] Some remarks on Wigner distribution, Theoret. Math. Phys. 17,

3,1163-1175 (1973).

Bergstrom, H. [Berg] On arrangements which determine limit theorems for eigenvalues

of a sample covariance matrix, Multivariate analysis Y(Proc. Fifth Internat. Sympos., Univ. Pittsburgh. 1978,201-212 North-Holland (1980).

Bessis, D., Itzykson, C. and Zuber, J. [BIZ] Quantum field theory techniques in graphical enumeration, Adv.

Appl. Math. 1, 109-157 (1980).

Bharucha-Reid, A. T. [Bha] Probabilistic Methods in Applied Mathematics. Academic Press,

(1970).

Bharucha-Reid, A. T. and Sambandham M. [BhaS] Random Polynomials. Academic Press, (1986).

References 469

Bogachev, L., Molchanov, L. and Pastur L. [BMP] On the density of states of random band matrices, Mat. Zametki,

50,31-42 (1991).

Bogomolny, E., Bohigas, O. and Leboeuf, P. [BBL1] Distribution of roots ofrandom polynomials, Phys. Rev. Lett., 68,

2726-2729 (1992).

Bogomolny, E., Bohigas, O. and Leboeuf, P. [BBL2] Quantum chaotic dynamics and random polynomials, Journal of

Statistical Physics, 85, Nos. 5/6,639-679 (1996).

Boutet de Monvel, A., Khorunzhy, A. and Vasilchuk, V. [BKV] Limiting eigenvalue distribution of random matrices with correlated

entries, Markov Processes Relat. Fields 2, 607-636, (1996).

Boutet de Monvel, A., Pastur, L. and Shcerbina, M. [BPS] On the statistical mechanics approach in the random matrix theory:

Integrated density of states, J. Stat. Phys. 79,3-4,585-613 (1995).

Brezin, E., Itzykson, C., Parisi, G., and Zuber, J. [BIPZ] Planar diagrams, Commun. Math. Phys. 59, 35-51 (1978).

Brezin, E. and Zee, A. [BrZ] Universality of the correlations between eigenvalues oflarge random

matrices, NucZ. Phys. B. 402,613-627 (1993).

Broder, A. [Bro] How hard is it to marry at random(on the approximation ofthe per­

manent), 19th Proceedings of the ACM Symposium on the Theory of Computing, 50-58 (1986).

Brody, T. A., Flores, J., French, J., Mello, P. A., Pandey, A., Wong, S. [BFF] Random-matrix physics: Spectrum and strength fluctuation, Rev.

Modern Phys. 58, 3, 385-479 (1981).

Bronk, B. V. [Brl] Accuracy of the semicircle approximation for the density of eigen­

values of random matrices, J. Math. Phys. 5, 215-220 (1964). [Br2] Exponential ensemble for random matrices, J. Math. Phys. 6,

228-237 (1965).

Brown, B. M. and Eagleson, G. K. [BrE] Martingale convergence to infinitely divisible laws with finite vari­

ances, Trans. Amer. Math. Soc. 162,2,449-453 (1971).

Burkholder, D. L. [Bur] Distribution function inequalities for martingales, Trans. Amer.

Math. Soc. 1, 1, 19-42 (1973).

Camarda H. S. and Georgopulos P. P. [eG] Statistical behaviour of atomic energy levels: agreement with ran­

dom matrix theory, Phys. Rev. Letters 50, 492-495 (1983).

470 References

Carrol, J., Ruppert, D. and Stefanski, L. A. [CRS] Measurement Error in Nonlinear Models. Chapman and Hall, NY,

305pp. (1995).

Casati G., Molinari, G. L. and Izrailev, F. M. [CMI] Scaling properties of band random matrices, Phys. Rev. Lett. 64,

1851-1854 (1980).

Chalker, J.T. and Mehlig, B. [ChM] Eigenvector statistics in non-Hermitian random matrix ensembles,

Probability Theory and Related Fields 93, 249-272 (1992).

Chan, T. [Ch] The Wigner semi-circle law and eigenvalues of matrix-valued dif­

fusions, Theoretical Physics 81, 16, 3367-3370 (1998).

Clarkson, K. L. [CIa] Las Vegas algorithms for linear and integer programming when

the dimension is smalL J. Assoc. Comput. Mach. 42, 2, 488-499(1995).

Cohen, J.E. and Newman, C.M. [CoN] The stability of large random matrices and their products, Ann.

Probab. 12, 283-310 (1984).

Constantinescu, F., Felder, G. M, Gawedzki, K. and Kupiainen, A. [CFGK] Analyticity of density of states, J. Stat. Phys. 48, 3, 365-391

(1987).

Crisanti, A., Paladin, G. and Vulpiani A. [Cra] Products of Random Matrices in Statistical Physics, Springer­

Verlag 88,113-131 (1993).

Crisanti, A. and Sompolinsky, H. [CS] Dynamics of spin systems with randomly asymmetric bonds:

Langevin dynamics and a spherical model, Phys. Rev. A. 36, 10, 4922-4939 (1987).

Curtiss, J. H. [Curt] Monte Carlo Method, National Bureau of Standards, Applied Math­

ematics Series 12 (1951).

Dagum, G. B. P., Mihail, M., Luby, M. and Vazirani, U. V. [DMLV] Poly tops, permanents, and graphs with large factors, Proceedings

of the 29th Annual IEEE Symposium on Foundations of Computer Science, 412-422 (1982).

Dauxois, J., Pousse, A. and Romain, Y. [DPR] Asymptotic theory for the principal component analysis of a vec­

tor random function: Some applications to statistical inference, J. Multivar. Anal. 12, 136-154 (1982).

References 471

Deev, A. D. [Dee] Representation of Discriminant Analysis Statistics and asymptotic

expansions when space dimensions are comparable with sample size. Soviet Math. Dokl. 951,759-762 (1970).

Deift, P., Kriecherbauer, T., Mclaughlin, K. T-R, Venakides, and Zhou, x.

[DKM1] Uniform Asymptotic for Polynomials Orthogonal with Respect to Varying Exponential Weights and Applications to Universality

Questions in Random Matrix Theory, Comunications on Pure and Applied Mathematics, Vol. LII, 1335-1425 (1999). [DKM2] Strong Asymptotics of Orthogonal Polynomials with Respect to

Exponential Weights, Comunications on Pure and Applied Mathe­matics, Vol. LII, 1491-1552 (1999).

Delyon, F., Simon, B. and Souillard, B. [DSS] From power pure point to continuous spectrum in disordered sys­

tems, Ann. Inst. H. Poincare, 42, 283-309 (1985).

Demetrefi, K. [Dem] Two-dimensional quantum gravity, matrix models and string the­

ory, Int. J. Mod. Phys. A. 8, 1185-1244 (1993).

Derrida, B. [Der] Random Energy Model; An exactly solvable model of disordered

systems, Phys. Rev. A. 24, 2613-2626 (1981).

Dochan, P. [Doch] Mixing Properties and Examples. Springer Verlag (1994).

Donoghue, W. [Don] Monotone Matrix Functions and Analytic Continuation. Springer

Verlag (1974).

Dyson, F. J. [Dys1] The dynamics of disordered linear chain, Phys. Rev. 92, 6, 1331-

1338 (1953). [Dys2] A Brownian-motion model for the eigenvalue of a random matrix,

J. Math. Phys. 3,6,31-60 (1962). [Dys3] Correlations between eigenvalues of a random matrix, Commun.

Math. Phys. 19,235-250 (1970).

Edelman, A. [Ed1] Eigenvalues and condition numbers of random matrices, SIAM 1.

Matrix Anal. Appl., 543-560 (1988). [Ed2] The distribution and moments of the smallest eigenvalue of a ran­

dom matrix of Wishart type, Lin. Alg. Appl. 159,55--80 (1991). [Ed3] On the distribution of a scaled condition number, Math. Comput.

58, 185-190 (1992).

472 References

[Ed4) The probability that a random real gaussian matrix has k real eigenvalues, related distribution and the circular law, Journal of Multivariate Analysis, 60, 203-232 (1997).

Edelman, A. and Kostlan, E. [EdK) How many roots of a random polynomial are real?, J. Amer. Math.

Soc. 32, 1-37 (1995).

Edelman, A., Kostlan, E. and Shub, M. [EKS] How many eigenvalues of a random matrix are real?, J. Amer.

Math. Soc. 7, I, 247-267 (1994).

Eltinge, J. L. [EL] Errors in variables estimation in the presence of serially correlated

observations, Contemporary Mathematics 112, 73-95 (1990).

Erdos, P. and Spencer, J. [ErS] The Probabilistic Method in Combinatorics. Academic Press, San

Diego, (1974).

Erdos, P. and Thran, P. [ErT] On the distribution of roots of polynomials. Ann. of Math. 51,

105-119 (1950).

Evangelou, S. N. and Katsanos, D. E. [EvK) Energy level statistics in disordered metals with an Anderson tran­

sition, Journal of Statistical Physics, 85, Nos. 5/6,525-550 (1996).

Faddeev, D. K. and Faddeeva, V. N. [FaF] Computational Methods in Linear Algebra. W.H. Freeman, San

Francisco (1963).

Farrel, R. H. [Far] Techniques of Multivariate Calculation. Lecture Notes on Mathe­

matics. Vol. 520, Springer Verlag(1976).

Feinberg J. and Zee, A. [FZ] Non-gaussian non-hermitan random matrix theory: Phase transi­

tion and addition formalism analysis, Nuclear Physics [FSJ. B 501, NO 3,643-669 (1997).

Fernandez, R., Frolich, J. and Sokal, A. [FFS] Random Walks, Critical Phenomena and Triviality in the Quantum

Field Theory. Springer Verlag, Heidelberg (1992).

Fine J. and Romain, Y. [FR) Reduced principal-component analysis, Math. Operationsforsch. u.

Statist. Ser. statist. 15, 493-512 (1984).

Forrester P. J. and Zuk J. A. [FG] Applications of the Dotsenko-Fateev integral in random-matrix

models, Nuclear Physics B 473, 616-630 (1966).

References 473

Fox, D. and Kahn P. B. [FK] Identity of the n-th order spacing distributions for a class of Hamil­

tonian unitary ensembles, Phys, Rev. 134, B1l51-B1l92 (1964).

Franchesco, P. D., Ginsparg, P. and Zinn-Justin, J. [FGZ] 2D gravity and random matrices. Phys. Rep. 254, (1) (1995).

Fuller W. A. [FUL] Prediction of true values for the measurement error model, Con­

temporary Mathematics 112,41-57 (1990).

Furedi, Z. and Komlos, J. [FK] The eigenvalues of random symmetric matrices, Combinatorica. 1,

2,233-241 (1981).

Fyodorov, Va. V. and Alexander D. M. [FA] Scaling properties of localization in random band matrices:

A s-model approach, Phys. Rev. Lett. 67, 2405-2409 (1991).

Fyodorov, Va. V. and Sommers, H. J. [FS] Statistics of resonance poles, phase shifts and time delays in quan­

tum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance, Journal of Mathematical Physics 38,4, 1918-1981 (1997).

Fyodorov, Va. V., Khoruzenko, B. and Sommers, H. J. [FKS] Almost-Hermitian random matrices: eigenvalue density in the com­

plex plane, Physics Letters A 226, 1-2,46-51 (1997).

Gaenssler, P., Strobel, J. and Stute, W. [GSS] On central limit theorems for martingale triangular arrays, Acta

Math. Acad. Sci. 3-4, 205-216 (1978).

Gantmacher, F. R. [Gan] The Theory of Matrices, vol. I, II. Chelsea, New York (1959).

Geman, S. A. [Gel] Limit theorem for the norm of random matrices, Ann. Probab. 8,

2,252-261 (1980). [Ge2] The spectral radius of large random matrices, Ann. Probab. 14,

1318-1328 (1986).

Geman, S. A. and Hwang, C. R. [GeH] A chaos hypothesis for some large systems of random equations. Z.

Wahrsch. verw. Gebiete 60, 291-314 (1982).

Ginibre, J. [Gin] Statistical ensembles of complex, quaternion, and real matrices, J.

Math. Phys. 6, 440-449 (1965).

Girko, V. L. [Girl] Limit theorems for permanent of random matrix, Theory Probab.

Math. Statist. 3, 29-34 (1971).

474

[Gir2]

[Gir3]

[Gir4]

[Gir5]

[Gir6]

References

Inequalities for a random determinant and a random permanent, Theory Probab. Math. Statist. 4, 42-50 (1971). Limit theorems for a random determinant I, Theory Probab. Math. Statist., 5,,25-31 (1971). Limit theorems for a random determinant II, Theory Probab. Math. Statist. 6,39-46 (1972). Refinement of some theorems for a random determinant and per­manent, Theory Probab. Math. Statist. 7, 28-32 (1972). Limit theorems for a random determinant. III, Theory Probab. Math. Statist. 8, 25-30 (1973).

[Gir7] Limit theorems for random quadratic forms. I, Theory Probab. Math. Statist. 9, 63-67 (1973).

[Gir8] Limit theorems for solution of system of linear random equations and the eigenvalues and determinant of random matrices, Sov. Math. Dokl. 14, 1508-1511 (1973).

[Gir9] Limit theorems for random quadratic forms. II, Theory Probab. Math. Statist. 10,47-54 (1974).

[Gir10] Eigenvalues of random matrices I, Theory Probab. Math. Statist. 11,8-14 (1974).

[Gir11] Limit theorems for eigenvalues of random matrices, Sov. Math. Dokl. 15,2,636-639 (1974).

[Gir12] Random Matrices. Kiev University, Kiev (1975). (in Russian). [Gir13] Limit theorems of general type for spectral functions of random

matrix, Theory Probab. Appl. 22,1,160-163 (1977). [Gir14] The central limit theorem for random determinants, Theory Probab.

Appl. 24,4,729-740 (1978). [Gir15] On the central limit theorem for random determinants, Theory

Probab. Math. Statist. 21, 35-39 (1979). [Gir16] Logarithmic law, Dokl. Akad. Nauk Ukr. SSR, Ser. Mat. 4, 35-39

(1979). (in Russian). [Gir17] Distribution of eigenvalues and eigenvectors of Hermitian random

matrices, Ukr. Mat. Zh., 31 (1979). [Gir18] On the uniqueness of the solution of the canonical spectral equation,

Ukr. Mat. Zh. 32, 6, 546-548 (1980). [Gir19] Theory of Random Determinants. Kiev University, Kiev (1980).

(in Russian). [Gir20] Stochastic Spectral Equation, in: Lecture Notes in Control and In­

formation Sciences, Stochastic Differential Systems, Filtering and Control, Springer Verlag (1980).

[Gir21] On new method of proving some limit theorems for product of in­dependent random matrices, Theory Probab. Math. Statist. 22 (1980).

[Gir22] The stochastic Ljapunov problem, Theory Probab. Math. Statist. 22 (1980).

References 475

[Gir23] The central limit theorem for random determinants, Theory Probab. Appl. 26, 3 (1981).

[Gir24] The polar decomposition of a random matrix, Theory Probab. Math. Statist. 23,21-31 (1981).

[Gir25] Limit theorems for determinants ofrandom Jacobi matrices, Theory Probab. Math. Statist. 24 (1982).

[Gir26] Distribution of eigenvalues and eigenvectors of unitary random ma­trices, Theory Probab. Math. Statist. 25, 13-16 (1982).

[Gir27] The V-transformation, Dokl. Akad. Nauk Ukr.SSR, Ser. Mat. 3, 5-6 (1982). (in Russian).

[Gir28] Eigenvalues of random matrices II, Theory Probab. Math. Statist. 27,31-32 (1983).

[Gir29] On the circle law, Theory Probab. Math. Statist. 28,15-23 (1983). [Gir30] Spectral theory of nonselfadjoint random matrices, Probability The­

ory and Mathematical Statistics, Proceedings of the USSR-Japan Symposium, Tbilisi, 1982, Lectures Notes Mathematics 1021, 153-156 (1983).

[Gir31] Central limit theorem for spectral functions of random Jacobi ma­trices, Theory Probab. Math. Statist. 29 (1984).

[Gir32] Regularized V-transformation, Theory Probab. Appl. 29,2, p.416-417 (1984).

[Gir33] Circular law, Theory Probab. Appl. 29,4,694-706 (1984). [Gir34] The elliptic law, Theory Probab. Appl. 30,4,640-651 (1985). [Gir35] Spectral theory of random matrices, Russian Math. Surveys 40, 1,

77-120 (1985). [Gir36] Distribution of eigenvalues and eigenvectors of orthogonal random

matrices, Ukr. Mat. Zh. 37, 5, (1985). [Gir37] Random Determinants, The Results of Science and Technology,

Theory of Probability and Mathematical Statistics, Theoretical Cy­bernetics, 24 (1986).

[Gir38] G2-estimates of spectral functions of covariance matrices, Theory Probab. Math. Statist. 35 (1986).

[Gir39] Introduction in general statistical analysis, Theory Probab. Appl. 32, 2, 250-265 (1987).

[Gir40] Elliptic law and elements of G-analysis, Probability Theory and Mathematical Statistics, Proceedings of the Fourth Vilnius Confer­ence, 1, VNU Science Press, Vilnius, 489-509 (1987). (in Russian).

[Gir41] Introduction in general statistical analysis, Proceedings of the First World Congress of the Bernoulli Society, vol. 2, 327-330 (1987).

[Gir42] Equation for Stieltjes transforms of limit spectral functions of iso­tropic random matrices, Theory Probab. Math. Statist. 36 (1987).

[Gir43] Two estimators of general statistical analysis, Theory Probab. Math. Statist. 37 (1987).

[Gir44] Multidimensional Statistical Analysis. Vyshcha Shkola, Kiev, (1988). (in Russian).

476

[Gir45]

[Gir46]

[Gir47]

[Gir48]

[Gir49]

[Gir50]

[Gir51]

[Gir52]

[Gir53]

[Gir54]

[Gir55]

[Gir56]

[Gir57]

[Gir58]

[Gir59]

[Gir60]

[Gir61]

[Gir62]

[Gir63]

[Gir64]

References

Spectral Theory of Random Matrices. Nauka, Moscow, (1988). (in Russian). Limit normalized spectral functions of self-adjoint random matrices pencil, Ukr. Mat. Zh. 40, 1, (1988). (in Russian). Asymptotic of the distribution of the spectrum of random matrices, Russian Math. Surveys 44, 4, 3-36 (1989). Limit theorems for distributions of eigenvalues of random symmet­ric matrices, Theory Probab. Math. Statist. 41, 23-29 (1989). Limit theorems for sums of distribution of eigenvalues of random symmetric matrices, Ukr. Mat. Zh. 41, 1, 23-30 (1989). (in Russian). G 25 estimator of principal components, Theory Probab. Math. Statist. 42, 1-10 (1990). G-estimates of eigenvalues of covariance matrices, Probability The­ory and Mathematical Statistics, Proceedings of the Fifth Vilnius Conference, Vol. 1, Vilnius (1990). (in Russian). Limit theorems for maximal and minimal eigenvalues of symmetric random matrices, Theory Probab. Appl. 35, 4, 677-690 (1990). G-consistent estimates of traces of matrix resolvents, eigenvalues, and eigenvectors, International Congress of Mathematicians, Ky­oto, Abstracts, Short Communications p. 149 (1990). Theory of Random Determinants. Kluwer Academic Publishers (1990). Theory of Empirical Systems of Equations. Lybid, Kiev (1990). (in Russian). Limit theorems for distributions of eigenvalues of random symmet­ric matrices, Theory Probab. Math. Statist. 41 (1990). G-consistent estimators for eigenvalues and eigenvectors of matrix, Factory's Laboratory 57,3 (1991). G-consistent estimators for eigenvalues and eigenvectors of matrix, Advances in Soviet Mathematics, Dynamical Systems and Statisti­cal Mechanics 3, 15-55 (1991). Asymptotic behavior of eigenvalues of empirical covariance matrices I, Theory Probab. Math. Statist. 44,37-44 (1992). Canonical spectral equation for singular spectral function of ran­dom matrices, Russian Math. Surveys 48,3,163-180 (1993). Asymptotic behavior of the eigenvalues of empirical covariance ma­trices 2, Theory Probab. Math. Statist. 45,36-42 (1993). On a class of Spectral Estimators. Theory of Probability and Math. Statist. 47, 15-22 (1993). Asymptotics of eigenvalues of symmetric random matrices, Math. Notices 54, 2, 769779 (1994). General equation for the eigenvalues of empirical covariance matri­ces I, Random Oper. Stoch. Eqs. 2, I, 13-25 (1994).

References 477

[Gir65] General equation for the eigenvalues of empirical covariance matri­ces II, Random Oper. Stoch. Eqs. 2, 2, 175-187 (1994).

[Gir66] The circular law: ten years later, Random Oper. Stoch. Eqs. 2, 3, 235-277 (1994).

[Gir67] The canonical equation for the resolvent of a random matrix with asymptotically independent entries I, Random Oper. Stoch. Eqs. 2,4,385-406 (1994).

[Gir68] Canonical equation for empirical covariance matrices, Proceedings of the 14th International Conference on Multidivariate Statistical Analysis, Warsaw 225-247 (1995).

[Gir69] Statistical Analysis of Observations of Increasing Dimension. Klu­wer Academic Publishers 286pp. (1995).

[Gir70] The canonical equation for the resolvent of a random matrix with asymptotically independent entries II, Random Oper. Stoch. Eqs. 3, 1,41-63 (1995).

[Gir71] The method of random determinants for estimating the permanent, Random Oper. Stoch. Eqs. 3,2,181-193 (1995).

[Gir72] The Elliptic law: ten years later I, Random Oper. Stoch. Eqs. 3, 3, 257-302 (1995).

[Gir73] The Elliptic law: ten years later II, Random Oper. Stoch. Eqs. 3, 4,377-398 (1995).

[Gir74] Canonical spectral equation for the eigenvalues of empirical covari­ance matrices, Proceedings of the 6th Vilnius Conference, Vilnius (1995).

[Gir75] Canonical spectral equation for the empirical covariance matrices. Ukrainian Math. Journal, 47, 9,1176-1189 (1995).

[Gir76] Random Matrices, Handbook of Algebra. Elsevier Science B.V., 27-78 (1995)

[Gir77] Spectral Theory of Minimax Estimation. Acta Applicandae Math­ematicae, 43, 59-69, (1996).

[Gir78] Limit theorems for permanent. Theory of Probability and Mathe­matical Statistics 53, 33-42 (1996).

[Gir79] The Canonical Spectral Equation. Theory of Probability and its Applications, 39, 4, 685-691 (1996).

[Gir80] Canonical Equation for the Resolvent of Empirical Covariance Ma­trix Random Operators and Stochastic Equations, 4, 1,61-76 (1996).

[Gir81] Strong Law for the eigenvalues and eigenvectors of empirical covari­ance matrices. Random Operators and Stochastic Equations, 4, 2, 179-204 (1996).

[Gir82] Strong Law for the eigenvalues of empirical covariance matrices. Multidimensional Statistical Analysis and Theory of Random Ma­trices. Proceedings of the Sixth Eugene Lukacs Symposium Held in Bowling Green State University, Department of Mathematics and

478 References

Statistics, Bowling Green Ohio. VSP Publishing, Utrecht, 61-92 (1996).

[Gir83] Distribution of spacings of random matrices. Part I: The First Spacing Law for Gaussian Hermitian matrices, Random Operators and Stochastic Equations 4, 3, 283-300 (1996).

[Gir84] Theory of Linear Algebraic Equations with Random Coefficients, Allerton Press, Inc. New York, 320pp. (1996).

[Gir85] A matrix equation for the resolvents of random matrices with in­dependent blocks. Theory of Probability and its Applications, 40, 4, 635-644 (1996).

[Gir86] Strong Law for singular values and eigenvectors of random matrices, 1. Random Operators and Stochastic Equations 5, 1,80-104 (1997).

[Gir87] Strong Circular Law. Random Operators and Stochastic Equations 5, 2, 173-197 (1997).

[Gir88] Ten years of general statistical analysis. (The main G-estimators of general statistical analysis). Relatorio de pesquisa, RP 43/97, Instituto de Matematica Estatistica e Computac;ao Cientifica, Uni­versidade Estadual de Campinas, Brasil, 97pp. (1997).

[Gir89] Simple proof of the strong circular law. Relatorio de pesquisa, RP 46/97, Instituto de Matematica Estatistica e Computac;ao Cien­tifica, Universidade Estadual de Campinas, Brasil, 22pp. (1997).

[Gir90] Convergence rate of the expected spectral functions of symmetric random matrices equals 0 (n -1/2)). Relatorio de pesquisa, RP 53/97, Instituto de Matematica Estatistica e Computac;ao Ci­entifica, Universidade Estadual de Campinas, Brasil, 53pp. (1997).

[Gir91] Strong Elliptic Law. Random Operators and Stochastic Equations, 5, 3, 269-306 (1997).

[Gir92] The V-density for eigenvalues of nonsymmetric random matrices and rigorous proof of the strong Circular law, Random Operators and Stochastic Equations, 5,4,371-406 (1997).

[Gir93] Refinement of the central limit theorem for random determinants, Theory of Probability and Its Applications, 42, I, 63-73 (1997).

[Gir94] Strong Law for the singular values and eigenvectors of random ma­trices II. Random Operators and Stochastic Equations, 6, N.3, 1998, 291-310.

[Gir95] Convergence rate of the expected spectral functions of symmetric random matrices equals O(n- l / 2 ). Random Operators and Stochas­tic Equations, 6, N.4, 1998, 359-406.

[Gir96] An Introduction to Statistical Analysis of Random Arrays. Pub­lishing VSP, 673pp.(1998).

[Gir97] The V -relation between Hermitian and non-Hermitian operators and strong law for normalized spectral functions of non-selfadjoint random matrices with independent row vectors. Markov Processes and Related Fields 4, 4, 499-508 (1998).

References 479

[Gir98] Strong Law for the singular values and eigenvectors of random ma­trices III. Inequalities for the spectral radius of large random ma­trices Random Operators and Stochastic Equations, 7, N.2, (1999), 179-200.

[Gir99]. Logarithmic law:twenty years later Theory of Probability and Mathematical Statistics, (Ukrainian) N.60, (1999), 17-25.

[Gir100] Strong law and canonical equation for normalized spectral functions of non self adjoint random Jacobi matrices Random Operators and Stochastic Equations, 8, N.1, 2000, 87-104 .

[Gir101] Random block matrix density and S5-Law Random Operators and Stochastic Equations, 8, N.2, 2000, 189-194 .

[Gir102] Twenty five years of stochastic canonical equation for normalized spectral functions of random symmetric matrices Random Opera­tors and Stochastic Equations, 8, N.3, 2000, 275-304 .

[Girl03] Stochastic canonical equation for normalized spectral functions of random Gram matrices Random Operators and Stochastic Equa­tions, 8, N.4, 2000, 397-313 .

[Gir104] Global Circular Law Random Operators and Stochastic Equations, 9, N.4, 2001, 400-406 .

Girko's Circular Law [GCL] Girko's Circular Law: Let A be Eigenvalues of a set of Random

n x n Matrices. Then AI Vn is uniformly distributed on the Disk. CRC Concise Encyclopedia of Mathematics on CD-ROM, 1996-9 Eric W. Weisstein

Girko, V. L. and Casati, G. [GiC1] Generalized Wigner law for band random matrices, Random Oper.

Stoch. Eqs_ 1, 1 (1993). [GiC2] Limit theorems for band random matrices whose entries have boun­

ded variances, Random Oper. Stoch. Eqs. 1, 2, 181-191 (1993).

Girko, V. L., Casati, G. and Molinari, L. [GCM] Equation for limit spectral functions of band random matrices, Ran­

dom Oper. Stoch. Eqs. 1, 1, 1-6 (1993).

Girko, V. L. and Dias, R [GiD] Numerical and Monte-Carlo verification of the V-Law, Random Op­

erators and Stochastic Equations, 6, 2, 201--212 (1998).

Girko, V. L., Ezhov, S. N. and Pluyko, V. A. [GEP1] Integral equations for average scattering matrix, Ukr. Fiz. Zh. 33,

10 (1988). (in Russian). [GEP2] The integral equations for the average stochastic scattering matrix,

Random Oper. Stoch. Eqs. 1,2, 161-170 (1993).

Girko, V. L. and Gupta, A. K. [GiG] Asymptotic behavior of spectral function of empirical covariance

matrices, Random Oper. Stoch. Eqs. 2, 1, 43-60 (1994).

480 References

Girko, V. L., Kirsch, W. and Kutzelnigg. A. [GKK] Necessary and sufficient conditions for the semicircle law, Random

Oper. Stoch. Eqs. 2, 2, 195-202 (1994).

Girko, V. L., Kokobinadze, T. S. and Chajka, O. G. [GDC] Distribution of the eigenvalues of Gaussian random matrices, Ukr.

Mat. Zh. 36, 1,9-12 (1984). (in Russian).

Girko, V. L. and Litvin, LN. [GiL] Stochastic Lyapunov problem for a system of stationary linear dif­

ferential equation, Differential Equations 24,8,1307-1324 (1988).

Girko, V. L., Olhovsky, V. S. and Chinarov, V. A. [GOC] To the theory of fluctuation of unitary S-matrix for nuclear reac­

tions, Izv. Akad. Nauk SSSR 48,1,166-171 (1984). (in Russian).

Girko, V. L. and Preston, N. [GiP] Numerical and Monte Carlo verification of the first spacing law,

Random Operators and Stochastic Equations 4, 4, 303-314 (1996).

Girko, V. L. and Repin, K. [GiR] One-Quarter-of-circumference Law, Theory of Probability and

Mathematical Statistics 50, 67-71 (1995).

Girko, V. L. and Rosen, D. [GiRo] Asymptotics for the normalized spectral function of matrix quadra­

tic forms, Random Oper. Stoch. Eqs. 2, 2, 153-162 (1994).

Girko, V. L. and Stepachno, 10 V. [GiS1] An estimator of Stieltjes transform of spectral function of singular

eigenvalues, Ukr. Mat. Zh. 4, 464-469 (1990). (in Russian). [GiS2] Asymptotics of Stieltjes transform of random spectral functions,

Theory Probab. Math. Statist. 46 (1993).

Girko, V. L. and Vasilev, V. V. [GiV1] Limit theorems for determinants of random Jacobi matrices, Theory

Probab. Math. Statist. 24 (1982). [GiV2] Limit theorems for the normalized spectral functions of nonselfad­

joint random Jacobian matrices, Theory Probab. Appl. 30, 1 (1985).

Girshick, M. A. [Girs] On the sampling theory of roots of determinantal equations, Ann.

Math. Statist. 10, 203-224 (1939).

Gnedenko, B. V. and Kolmogorov, A. N. [GnK] Limit Distribution for Sums of Independent Random Variables.

Addison-Wesley, New York (1954).

Gold'sheid, Ya. and Guvarc'h, Y. [GG] Zariski closure and the dimension of the Gaussian law of the prod­

uct of random matrices. I, Probab. Theory Relat. Fields 105, 5, 109-142 (1996).

References 481

Gold'sheid, Ya. and Khoruzhenko, B. A. [GK] Distribution of eigenvalues in Non-Hermitian Anderson models,

Phisical Review Letters, 80, 13, 2897-2900 (1998).

Gold'sheid, Ya. and Margulis, G. A. [GoM] Lyapunov indices of a product of random matrices, Russian Math.

Surveys 44,5, 11-71 (1989).

Grenander, U. [Gre] Probabilities on Algebraic Structures. Wiley, New York-London

(1963).

Grenander, U. and Szego, G. [GrS] Toeplitz Forms and Their Applications. University of California

Press, Berkeley, California (1958).

Grenander, U. and Silverstain, J. W. [GrSil] Spectral analysis of networks with random topologies SIAM J.

Appl. Math., 32, 2, 499-519 (1977).

Gudmundsson, T., Kenney, C. and Laub, A. J. [GKL] Small-sample statistical estimates for the sensitivity of eigenvalue

problem, SIAM Journal on Matrix Analysis and Applications 18, 4, (1997).

Guhr, T. [Guh] Transitions toward quantum chaos: with supersymmetry from Pois­

son to Gauss, Annals of Physics 250, 145-192 (1996).

Guhr, T. and Wettig T. [GW] An Itzykson-Zuber-like integral and diffusion for complex ordinary

and supermatrices, J. Math. Phys. 37, 12,6395-6413 (1996).

Haake, F. [Haa] Quantum Signatures of Chaos. Springer Verlag, Heidelberg (1991).

Hammersley, J. M. [Ham] The zeros of a random polynomial, Proceedings of the Third Berke­

ley Symposium on Mathematical Statistics and Probability, Univer­sity of California Press, Berkeley-Los Angeles, 2, 89-111 (1956).

Hatano, N. and Nelson, D.R. [HaNl] Localization transitions in non-Hermitian quantum mechanics,

Phys. Rev. Lett. 77, 570-573 (1996).

Hatano, N. and Nelson, D.R. [HaN2] Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev.

B56, 8651-8673 (1997).

Hasegawa, H. [Has] Density of states in aperiodic solids, Supplement of the Progress of

Theoretical Physics 53 (1973).

482 References

Hertz, J. A. [Her] Statistical dynamics of learning, in: Lecture Notes in Physics, (Ed.

Luis Garrido), Statistical Mechanics of Neural Networks, Proceed­ings, Sitges, Barcelona, Spain, Springer-Velag (1990).

Hotelling, H. [Hot] Analysis of a complex of statistical variables into principal compo­

nents, J. Educ. Psychol. 24, 417-441, 498-520 (1933).

Hwang, C. R. [Hwa] A brief survey on the spectral radius and spectral distribution of

large random matrices with i.i.d. entries. In Random Matrices and their Applications. Contemporary Mathematics 50, 145-152 (1986).

Ibragimov, I. A. and Linnik, Yu. V. [IbL] Independent and Stationary Random Variables, Wolters-Noordhoff,

Groningen (1971).

Ibragimov, I. A. and Maslova , N. B. [IbM] On the expected number of real zeros of random polynomials. I.

Ishii, K.

Coefficients wth zero means, Theory of Probab. and Its Applications 16,229-248 (1971).

[Ish] Localization of Eigenstates and Transport Phenomena in the One­Dimensi- onal Disordered System, Supplement of the Progress of Theoretical Physics, 53 (1973).

Iso, S. and Kavalov, A. [IK] Breakdown of universality in random matrix models, Nuclear Phy­

sics [FSj. B 501, NO 3, 670-684 (1997).

Itzykson, C. and Zuber, J. B. [ItZ] The planar approximation II, J. Math. Phys. 21, 3, 411-421

(1980).

Izrailev, F. M. [Izr] Simple models of quantum chaos: spectrum and eigenfunctions,

Phys. Report. 196, N 5-6, 299-392 (1990).

James, A. T. [Jam] The non-central Wishart distribution, Proc. Roy. Soc. London

229A, 1178,364-366 (1955).

Janik, R. A., Nowak, M. A., Papp, G. and Zahed, I. [JNPZ] Non-hermitan random matrix models, Nuclear Physics [FSj.

B 501, NO 3, 603-642 (1997).

Jerrum, M. R. and Sinclair, A. J. [JeS] Approximating the permanent, SIAM J. Comput. 18, 1149-1178

(1989).

References 483

Johansson, K. [Joh] On fluctuations of eigenvalues of a random Hermitian matrices,

Duke Mathematical Journal. 91, N1, 151-204 (1998).

Jonsson, D. [Jon] Some limit theorems for the eigenvalues of a sample covariance

matrix, J. Multivar. Anal. 12, 1-38 (1982).

Juhasz F. [Ju1] On the asymptotic behaviour of the spectra of non-symmetric ran­

dom (0,1) matrices, Discrete Math. 41, 161-165 (1982). [Ju2] On the characteristic values of non symmetric block random matri­

ces, J. Theoret. Probab. 3, 2, 199-205 (1990). [Ju3] On the eigenvalues of block random matrices, Linear Algebra and

its Applications, 246, 225-231 (1996). [Ju4] On the spectrum of a random graph. In Algebraic and Discrete

Methods in Graph Theory (L. Lovasz and V.T. Sos, Eds.) Proc. ColI. Math. Soc. J. Bolyai 25, I, 313-316 (1981).

Kac, M. [Kac] On the average number of real roots of random algebraic equation,

Bull. Amer. Math. Soc. 49, 314-320 (1943).

Kaijeser T. [Kai] A note on random systems with complete connection and their

applications to products of random matrices, Contemporary Math­ematics, Random Matrices and Their Applications, 50, 243-254 (1986).

Karabali, D., Sakita, B. [KS] Semiclassical approach for nonrelativistic fermions in low dimen­

sions, International Journal of Modern Physics, A, 6,28,5079-5097 (1991).

Karmarkar, N., Karr, R., Lipton, R., Lovasz, L. and Luby, M. [KKLLL] A Monte Carlo algorithm for estimating the permanent, SIAM J.

Comput. 22,2,284-293 (1993).

Karp, R. M., Luby, M. and Mardas, N. [KLM] Monte Carlo approximation algorithms for enumeration problems,

J. Algorithms 10, 429-448 (1989).

Kasin, B. S. [Kas] On the mean value of certain function connected with the conver­

gence of orthogonal series, Anal. Math. 4, 27-35 (1978).

Kato, T. [Kat] Perturbation Theory for Linear Operators. Springer Verlag, Berlin

(1966).

Kellogg, O. D. [Kel] Foundations of the Potential Theory. Dover, New York (1953).

484 References

Khorunzhy, A. and Pastur, L. [KP] Limits of infinite interaction radius, dimensionality and the number

of components for random operators with off-diagonal randomness, Commun. Math. Phys. 153, 605-646 (1993).

Khorunzhy, A., Khoruzhenko, B. and Pastur, L. [KKP1] On the liN corrections to the Green functions of random matri­

ces with independent entries, J. of Physics A. Mathematical and General. 28, L31-L35 (1995).

[KKP2] Asymptotic properties of large random matrices with independent entries, J. Math. Phys. 37, 10, 5033-5060 (1996).

Khorunzhy, A., Molchanov, S. and Pastur L. [KMP] On the eigenvalue distribution ofband random matrices in the limit

of their infinite order, Teor. Mat. Fiz. 90, 163-178 (1992).

Kollo, T. and Neudecker, H. [KoN1] Asymptotics of eigenvalues and unit-length eigenvectors of sample

variance and correlation matrices, J. Multivar. Anal. 47, 283-300 (1993).

Kollo, T. and Neudecker, H. [KoN2] Asymptotics of eigenvalue-normed eigenvectors of sample variance

and correlation matrices, Proceedings of the sixth Lukacs Sympo­sium, 111-128 (1996).

Komlos, J. [KomI] On the determinant of (0, 1) matrices, Studia Scient. Math. Hung.

3,4 (1967). [Kom2] On the determinant of random matrices, Studia Scient. Math.

Hung. 2, 1-2 (1968).

Konishi, S. [Koni] Asymptotic expansions for the distributions of statistics based on

the sample correlation matrix in principal-component analysis, Hi­roshima Math. J. 9, 647-700 (1979).

Konyagin, S. V. [Kon] On minimum of modulus of random trigonometric polynomials with

coefficients ±1, Mathematical Notes 56,80-101 (1994).

Kotani, S. [Kot] Lyapunov exponents and spectra for one-dimensional random

Schrodinger operators, Comtemporary Mathematics, Random Ma­trices and Their Applications, 50, 277-286 (1986).

Kuz, M., Lewentein, M. and Haake, F. [KLH] Density of eigenvalues of random band matrices, Phys. Rev. A 44,

5,2800-2808 (1991).

Landkof, N. S. [Lan] Foundations of Modern Potential Theory, Springer-Verlag, 425p.

(1972).

References 485

Larsen, F. [Lar] Brown measures and R-diagonal elements in finite von Neumann

algebras, Ph.D. thesis, University of Southern Denmark (1999).

Lehmann, N. and Sommers, H. J. [LeS] Eigenvalue statistics of random real matrices, Phys. Rev. Lett. 67,

8, 941~944 (1991).

Lifshits, I. M. [Lif] On the structure of the energy spectrum and quantum state of

disordered condenser systems, Sov. Fiz. Usp. 7, 549~573 (1965). (in Russian).

Lovasz, L. [Lov] On determinants, matchings, and random algorithms, Budach, L.

(editor), Fundamentals of Computing Theory, Akademica Verlag, Berlin (1979).

Lovasz, L. and Plummer, M. D. [LoP] Matching Theory. Academic Press, New York, 1986.

Lubinsky, D. S. [Lub] Strong asymptotics for extremal errors and polynomials associated

with Erdos-type weights. Longman Group UK Limited, 240p., 1989.

Luby, M., Sinclair, A., and Zuckerman, D. [LSZ] Optimal speedup of Las Vegas algorithms, Inf. Process. Lett. 47,

173~ 180 (1993).

Marcus, M. and Minc, H. [MaM] A Survey of Matrix Theory and Matrix Inequalities. Allyn and

Bacon (1964).

Marchenko, V. A. and Pastur, L. A. [MaP] Distribution of the eigenvalues in certain sets of random matrices,

Mat. Sb. 1, 457~483 (1967). (in Russian).

Mardia, K. V., Kent, J. T. and Bibby, J. M. [MKB] Multivariate Analysis. Academic Press, London (1979).

Marriot, F. M. [MaM] Practical problems in a method of cluster analysis, Biometrics 27,

501~514 (1971).

Megrabian, R. M. [Meg] On a characteristic of random matrices connected with uncondi­

tional convergence almost everywhere, Anal. Math. 14, 37~47 (1988).

Mehta, M. L. [Mehl] Random Matrices and the Statistical Theory of Energy Levels. Aca­

demic Press, New York-London (1967). [Meh2] Random matrices. Academic Press, San Diego (1991).

486 References

Memon, A. Z. and Okamoto., M. [MO] . Asymptotic expansion of the distribution of the z statistic in

discriminant analysis. J. Multivariate Analysis 1 294-307 (1971).

Mezinescu G. A., Bessis, D., Fournier, J. D., Mantica, G. and Aaron, F. D.

[MBF] Distribution of Roots of Random Real Generalized Polynomials. Journal of Statistical Physics, 86, 3/4, 59-71 (1997).

Miller, S. M. [Mil] Analysis of residuals from measurement error models, Contempo­

rary Mathematics, 112,59-71 (1990).

Minc, H. [Min] Permanents. Encyclopedia of Mathematics and its Applications, 6.

Addison-Wesley, Reading (1978).

Mizutori, S. and Zelevinsky, V. G. [MZ] Level and width statistics for a decaying chaotic system, Z. Phys.

A 346, 1-9 (1993).

Molchanov, S. A., Pastur, L. A. and Khorunzhy, A. M. [MPK] Distribution of eigenvalues of random band matrices in the limit of

infinite order, Teor. Mat. Fiz. 90, 2 (1992). (in Russian).

Moro, J., Burke, J. V. and Overton, M. L. [MBO] On the Lidskii-Vishik-Lyusternik Perturbation Theory for Eigen­

values of Matrices with Arbitrary Jordan Structure, SIAM Journal on Matrix Analysis and Applications 18, 4, (1997).

Muirhead, R. J. [Muir] Aspects of Multivariate Statistical Theory. Wiley & Sons, New York

(1982).

Muskhelishvili, N. I. [Musl] Singular Integral Equations. Noordhoff, Groningen (1953). [Mus2] Some Principal Problems of Mathematical Theory of Elasticity.

Nauka, Moscow (1966). (in Russian).

Nelson, David R. and Shnerb, Nadav M. [NS] Non-Hermitian localization and population biology, Phys. Rev. E

(3) 58,2, part A, 1383-1403 (1998).

Neudecker, H. and Wesselman, A. M. [NW] The asymptotic variance matrix of the sample correlation matrix,

Linear Algebra Appl. 127, 589-599 (1990).

Nica, A., and Speicher, R. [NSl] R-diagonal pairs-a common approach to Haar unit aries and circu­

lar elements, Free probability theory (Waterloo, ON, 1995), Fields lnst. Commun., vol. 12, Amer. Math. Soc., Providence, Rl, 149-188 (1997).

References 487

[NS2] Commutators of free random variables, Duke Math. J. 92, 3, 553-592 (1998).

Nuquist, H., Rice, S., and Riordan, O. [NRR] The distribution ofrandom determinants, Quart. Appl. Math. 12,

2 (1954).

Okamoto, M. [0] An asymptotic expansion of the distribution of linear discriminant

function. Ann. Math. Stat. 34, 1286-1301 (1963).

Olver, F. W. J. [Olv] Asymptotics and Special Functions. Academic Press, New York

(1974).

Opperman, R. and Wegner, F. [OpW] Disordered system with n orbitals per site: n - 1 expansion, Z.

Phys. 34,327-348 (1979).

Paige, C. C. and Saunders, M. A. [PaS] Solutions of sparse indefinite systems of linear equations, SIAM J.

Numer. Anal. 12,617-629 (1975).

Papadimitriou, C. H. [Pap] Complexity Theory. Addison-Wesley, Reading, 1994.

Pastur, L. A. [Pas1] Spectra of random self-adjoint operators, Russian Math. Surveys

28, 1,4-63 (1973). [Pas2] Spectral properties of disordered systems in the one-body approxi­

mation, Commun. Math. Phys. 75,179-196 (1980). [Pas3] On the universality of the level spacing distribution for some en­

semble of random matrices, Lett. Math. Phys. 25, 259-265 (1992).

Pastur, L. A. and Vasilchuk, V. [Pa V] On the law of additional of random matrices, Comm. Math. Phys.

214, 2, 249-286 (2000).

Pastur, L. A. and Shcherbina, M. [PaS] Universality of the local eigenvalue statistics for a class of unitary

invariant random matrix ensembles, J. Stat. Phys. 86, 109-147 (1997).

Pastur, L. A. and Figotin, A. [PaF] Spectra of Random and Almost Periodic Operators. Springer Verlag

(1992).

Pastur, L. A. and Khorunzhy, A. M. [PaKJ Limit Distribution of Eigenvalues of Random Operators of Wegner

Model, Preprint FTINT 389, Kharkov (1989). (in Russian).

Petrov, V. V. [Pel] Sums of Independent Random Variables. Springer -Verlag, 346pp.

(1975).

488 References

Porter, C. E. [Por] Statistical Theories of Spectra: Fluctuations, Collection of Reprints

and Original Papers, Academic Press, New York (1965).

Portnoy S. [Portl] Asymptotic behavior of M -estimators of p regression parameters

when p2 In is large, Ann. Statist. 12, 1298-1309 (1984). [Port2] Asymptotic behavior of M-estimators of p regression parameters

when p2 In is large. II. Normal approximation, Ann. Statist. 13, 1403-1417 (1985).

[Port3] Asymptotic behavior of the empirical distribution of M-estimated residuals from a regression model with many parameters, Ann. Statist. 14, 1152-1170 (1986).

[Port4] On the CLT in RP when p --t 00, Probab. Theory Related Fields 73, 571-584 (1986).

Poschel, J. [Pos] Examples of discrete Schrodinger operators with pure point spec­

trum, Commun. Math. Phys. 88, 447-463 (1983).

Prekopa, A. [Pre] On random determinants I, Studia Scient. Math. Hung. 2, 1-2

(1967).

Preston, N. [Pres] On Girko's Cubic Law, Random Operators and Stochastic Equa­

tions. 6, 1,89-94 (1998).

Reed, M. and Simon, B. [RS] Methods of Modern Mathematical Physics. IV: Analysis of Opera­

tors. Acad. Press. London (1978).

Rice, S. O. [Ric] Mathematical theory of random noise, Bell System Tech. J. 25,

46-156 (1945).

Ryan, 0. [Ry] On the limit distributions of random matrices with independent or

free entries, Commun. Math. Phys. 193, No 3 595-626(1998).

Schafer L. and Wegner F. [Sch W] Disordered system with n orbitals per site: Lagrange formulation,

Z. Phys. 38, 113-126 (1980).

Schmidt, H. [Sch] Disordered one-dimensional crystals, Phys. Rev. 105, 2, 425-441

(1957).

Selberg, A. A. [Sell Remarks on a multiple integral. (Norwegian) Bemerkninger om et

multippel integral, Norsk Matematisk Tidsskrift 26, 71-78 (1944).

References 489

Serdobolsky, V.1. [Serl] On classification errors induced by sampling, Theory of Probability

and its Applications 24, 130-143 (1979). [Ser2] Resolvent and spectral functions of sample covariance matrices of

increasing dimension, Russian Math. Surveys 40, (1985).

Shepp, L. A. and Vanderbei R.J. [SV] The complex zeros of random polynomials, Transactions of the

American Mathematical Society 347, 11,4365-4384 (1995).

Sherington, D. and Kirkpatrick, S. [SK] Solvable model of a spin glass, Phys. Rev. Lett. 35, 1792-1796

(1975).

Silverstein, J. W. [Sill] The smallest eigenvalue of a large-dimensional Wishart matrix,

Ann. Probab. 13,1364-1368 (1985). [Si12] On the weak limit of the largest eigenvalue of a large-dimensional

sample covariance matrix, J. Multivar. Anal. 30,307-311 (1989). [Sil3] Eigenvalues and eigenvectors of large dimensional sample covari­

ance matrices, Contemporary Mathematics, Random Matrices and Their Applications, 50, 153-159 (1986).

Silverstein, J. W. and Bai, Z. D. [SiB] On the empirical distribution of eigenvalues of a class of large

dimensional random matrices, J. Multivar. Anal. 54, 175-192 (1995).

Silverstein, J. W. and Choi, S. I. [SiC] Analysis of the limiting spectral distribution of large dimensional

matrices, J. Multivar. Anal. 54, 295-309 (1995).

Silverstein, J. W. and Combettes, P. L. [SiCo] Signal detection via spectral theory of large dimensional random

matrices, IEEE Truns. Signal Processing 40,2100-2105 (1995).

Sinai, Ya. G. and Soshnikov, A. [SiS] A refinement of Wigner's semicircle law in a neighborhood of the

spectrum edge for random symmetric matrices, Functional Analysis and its Applications 60,114-131 (1998).

Sinclair, A. J. [Sin] Algorithms for Random Generation and Counting: A Markov

Chain Approach. Birkhauser, Basel (1993).

Sommers, H. J., Crisanti, A., Sompolinsky, H. and Stein, Y. [SCSS] Spectrum of large random asymmetric matrices, Phys. Rev. Lett.

60, 1895-1898 (1988).

Soshnikov, A. [Sosl] Level spacings distribution for large random matrices: Gaussian

fluctuations, Annals of Mathematics 60, 573-617 (1998).

490 References

Sparo, D. I. and Sur, M. G.

Stein, C.

[55] On the distribution of roots of random polynomials, Vestnik M os­kov. Univ. Ser. I Mat. Mekh. 112, 40-53 (1962).

[5tl] Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proc. 3rd Berkeley Symp. Math. Stat. Prob. 1, 197-206 (1955).

[5t2] The admissibility of Hotellings T2 test, Ann. Math. Stat. 27, 616-623 (1956).

[5t3] Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Inst. Math. 16, 155-160 (1964).

Stockmann, H. J. and Seba, P. [55] The joint energy distribution function for the Hamiltonian H =

Ho- C WW+ for the one-channel case, J. Phys. A: Math. Gen. 31,No 15,3439-3448 (1998).

Stone, A. D., Mello, P. A., Muttalib, K. A. and Pichard, J. L. [5MMP] Random theory and maximum entropy models for disordered con­

ductors. In Mesoscopic Phenomena in Solids. 56, eds. B.L. Altshuler, P.A. Lee, and R.A. Webb, North-Holland, Amsterdam, Ch.9, 369-448 (1991).

Sugiura, N. [5ug] Asymptotic expansions of the distributions of the latent roots and

latent vectors of the Wishart and multivariate F-matrices, J. Mul­tivar. Anal. 6, 500-525 (1976).

Szego, G. [5ze] Orthogonal Polynomials, American Mathematical Society, Provi­

dence, (1939).

Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., Yagola, A. G. [TG5Y] Numerical Methods for the Solution of Ill-Posed Problems, Kluwer,

252p. (1995).

Tracy, A. C. and Widom, H. [TrW1] Level spacing distributions and the Airy kernel, Commun. Math.

Phys. 159,151-174 (1994). [TrW2] Level spacing distributions and the Bessel kernel, Commun. Math.

Phys. 161, 289-309 (1994). [TrW3] Fredholm determinants, differential equations and matrix models,

Commun. Math. Phys. 163, 33-72 (1994). [TrW4] On orthogonal and symplectic matrix ensembles, Commun. Math.

Phys. 177, 727-754 (1996).

Tyler, D. [Tyll] Asymptotic inference for eigenvectors, Ann. Statist. 9, 725-736

(1981).

References 491

[TyI2] Robustness and efficiency properties of scatter matrices, Biometrika 70,411-420 (1983).

Valiant, L. G. [Vall] The complexity of computing the permanent, Theor. Compo Sci.

8, 189-201 (1979). [VaI2] The complexity of enumeration and reliability problems, SIAM J.

Comput. 8,410-421 (1979).

Vere-Jones, D. rVer] Alpha-permanents and their applications to multivariate Gamma,

negative binomial and ordinary binomial distributions, New Zealand Journal of Mathematics 26,125-149 (1997).

Vilenkin, N. Ja. [ViI] Special Functions and Theory of Group Representations, American

Mathematical Society, Providence (1968).

Voiculescu, D. V., Dykema, K. J., and Nica, A. [VDN] Free random variables. American Mathematical Society, Provi­

dence, RI (1992).

Wachter, K. W. [Wac] The strong limits of random matrix spectra for sample matrices of

independent elements, Ann. Probab. 6, 1-18 (1978).

Wald, A. [Wall Estimation of a parameter when the number of unknown param­

eters increases indefinitely with the number of observations, Ann. Math. Statist. 19,2, 220-227 (1948)

Watson, G. N. [Wat] A Treatise on the Theory of Bessel Functions, Cambridge, 1922.

Wegmann, R. [We] The asymptotic eigenvalue-distribution for a certain class of random

matrices, J. Math. Anal. Appl. 56, 113-132 (1976).

Wegner, F. [Weg] Disordered system with n orbitals per site: n = 00 limit, Phys.

Rev. B. 19, 2, 783-792 (1979)

Whittermore A. S. [WH] Errors-in-variables regression problems in epidemiology, Contemp.

Mathematics 112,17-31 (1990).

Wigner, E. P. [Wigl] Statistical properties of real symmetric matrices with many dimen­

sions, Can. Math. Proc., 174-198 (1957). [Wig2] On the distribution of the roots of certain symmetric matrices, Ann.

Math. 67,2,325-327 (1958). [Wig3] Random matrices in physics, SIAM Rev. 9, 1, 1-23 (1967).

492 References

Wilkins, J. E. jun. [Wilk] The expected value of the number of real zeros of random sum

of Legendre polynomials. Proc. Am. Math. 125, 5, 1531-1536 (1997).

Wilkinson, J. H. [Will The Algebraic Eigenvalue Problem, Clarendon Press, Oxford

(1965).

Wilkinson, J. H. and Reinsch, G. tWiRl Algebra. Springer Verlag (1971).

Yin, Y. Q. [Yin] Limiting spectral distribution for a class of random matrices, J.

Multivar. Anal. 20, 50-68 (1986).

Yin, Y. Q., Bai, Z. D. [YB] Spectra for large-dimensional random matrices, Contemporary

mathematics, Random Matrices and Their Applications, 50, 161-167 (1986).

Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. [YBK] On limit of the largest eigenvalue of the large-dimensional sample

covariance matrix, Tech. Report No. 84-44, Pittsburgh University, Pittsburgh (1984).

Yin, Y. Q. and Krishnaiah, P. R. [YiK1] A limit theorem for the eigenvalues of product of two random ma­

trices, J. Multivar. Anal. 13, 489-507 (1984). [YiK2] Limit theorem for the eigenvalues of the sample covariance matrix

when the underlying distribution is isotropic, Theory Probab. Appl. 30,861-867 (1985).

[YiK3] On the limit of the largest eigenvalue of the large-dimensional sam­ple covariance matrix, Probab. Theory Ret Fields 78, 509-521 (1988).

Zakrewski, J., Dupret, K. and Delande, D. [ZDD] Wigner or non-Wigner: that is the question, Proceedings of the

XXXIst Winter School of Theoretical Physics "Chaos-The Interplay Between Stochastic and Deterministic Behavior" , Springer Verlag, 559-564, (1995).

Index

A Algebra

a -, 1, 6, 13, 20, 66, 191, 194, 195,234,241,246,327,337, 371, 389, 409, 455

Analysis General Statistical -, 206

Approximation Stirling's -, 338, 372

Assertion G -, 225

C Chain

Markov-,438 Coefficient

asymptotic independence -,412, 455, 461

Condition block asymptotic independence

-, 409, 454 doubly stochastic -, 117, 123,

131 of the uniform asymptotic neg­

ligibility -, 51, 183 G -, 205, 389, 392, 397, 403,

404,414,415,423,441,446 Lindeberg -, 2, 3, 10, 11, 23,

45,97,106,117,119,124, 126, 134, 140, 142, 147, 150, 184,210,253,259,305,378, 380

Lindeberg modified -, 25, 43, 49, 132

Ljapunov's -, 305 M -, 423, 461

Continuation analytic -, 112

493

Convergence weak -, 52, 77

D Determinant, 128, 168, 407

Fredholm -, 183 Distribution

Bronk-Marchenko-Pastur-,132 Cauchy -, 297 Circular -, 231 limit spectral -, 117 normal-, 96, 225, 291, 305, 306,

308, 350, 394 multidimensional -, 55 two-dimensional -, 1

Density block matrix -, 426, 429 Bronk-Marchenko-Pastur(BMP)

-, 132, 134, 151, 152 Cauchy -, 297 Cubic -, 130 Sombrero probability -, 396, 463,

464 spectral -, 126 V-,324, 357,366,381, 383,387

E Eigenvalue

extreme -, 184 Eigenvector 19 Entries

infinitely small -, 153 pairs -,323 stationary -, 431

Equation accompanying -, 36, 142 Berezin's -, 87 Cardano -, 49 differential -, 226

494

higher orders -, 227 G -,228 Ll -,36,37, 139, 141, 274 L4 -, 274 Ml -,35 stable -, 95 stable stochastic -, 96 quadratic -, 45 regularized stochastic -, 68, 70 Ricatti -, 426 spectral stochastic -, 75, 85 V -, 379, 381, 384, 385

Equation canonical Kl -, 3, 14, 16, 19, 24, 424 K2 -, 26, 41, 42 K3 -,52 K4 -, 76, 86, 89 K5 -, 87, 89 K6 -, 94, 95 K7 -, 113, 118, 125 Kg -, 120, 122-124 K9 -,126 KlO -, 129, 130 Kn -, 142, 145, 150 K12 -, 154, 159 K 13 -, 162, 363 K14 -, 166, 187, 189 K 15 -, 181, 182 K 16 -, 186, 187, 189 K17 -, 203, 204, 208, 210, 211; K 18 -, 208, 209, 211 K 19 -, 228 K 20 -, 253 K21 -, 261, 262, 269, 271, 276,

308 K22 -, 324, 349, 351, 354 K 23 -, 356, 358 K 2C, 360, 362, 364 K 25 -, 380 K 26 -, 384, 386, 392, 397, 399,

400 K27 -, 404, 417, 418, 419, 423 K28 -, 425, 427, 432, 433, 434 K 29 -, 434, 436, 437, 438 K30 -, 439

Estimator

Index

G -,229 G2 -, 205 G19 -, 230 modified G2 -, 206

F Form

quadratic random -, 8, 157, 158 Formula

Levy -, 162 Levy-Khintchine -, 76 Cardano -, 50, 152, 243, 255,

256 hermitian -, 264 inverse -, 82 inverse for V-transform -, 233 matrix perturbation -, 3 random matrix perturbation -

218,219, Sterling's - 311, perturbation of resolvent -,139,

157, 159 Function

Borel- 227, bounded variation -, 208 characteristic -, 57, 77 degenerate spectral -, 82 Dirac delta -, 150 G - 261, 263, 264, 276, individual -, 40, 88 normalized spectral -, 4, 5, 14,

17, 47, 77, 80, 126, 127, 128,141,153,165,205,208, 210,232,233,260,262,322, 325,349,354,359,363,385, 388,390,397,401,402,428, 436, 438, 453

spectral -, 18, 52, 77, 80, 82, 83, 91, 145, 183, 205, 222, 223,232,243,252,263,314, 322, 344, 378, 396, 443

strength -, 18 Functional 76

degenerate linear -, 87 I Independence

block asymptotic -, 409

Index

Inequality Berry~Esseen~, 243,390 Burkholder ~, 7, 21, 191, 239,

329 Cauchy~Schwarz~Bunhakowski

~,9, 11, 106, 107,413,414, 415

martingale differences ~, 371 L5 ~, 274

Integral

L Law

Fourier ~, 298

Cauchy ~, 297 Circular ~, 260 Cubic ~, 26, 131 Elliptic ~, 262, 266, 316, 321 normal ~, 14, 34, 26, 61, 172,

214,221,228,296,310,417 random infinitely divisible~, 57,

65 quarter~of~circus ference~, 150 Semicircular ~, 26, 41, 47, 89,

268, 278, 403, 426, 429 SS~, 429 stable ~, 93 standard Normal ~, 170, 201,

260, 268, 269, 276, 322 Strong ~, 4, 19 Strong Circular ~, 243, 253 Strong Elliptic ~, 267 Strong V ~, 380 V~, 378 Weak~, 22

Lemma Borel~Cantelli~, 8, 22, 191, 239,

330 Limit theorem

central ~, 304 Logarithm

normalized ~, 180 M Markov chain, 438 Martingale-differences, 13, 239, 246 Matrix

ACE ~, 51, 68, 69

495

block~, 196,351, 401, 403~406, 419, 420, 428, 440, 454

complex~, 5 covariance~, 141, 150, 151, 186,

187,188,205,206,208,210, 228,305

double--stochastic~, 168, 180,379 diagonal~, 26, 124, 379, 385, 394 empirical covariance~, 141, 188,

152, 208 G ~, 274, 315, 349, 359, 361,

363, 390, 392 Gaussian Gram ~, 128 Gaussian nonsymmetric ~, 291 Gram ~, 27, 165, 168, 183, 244,

312, 332, 368 Hermitian ~, 4, 5, 97, 131, 156,

232, 243, 263, 264, 380 Hermitian Gaussian ~, 291 idempotent ~, 171 inverse ~, 171 nonnegative definite~, 62 nonhermitian~, 97 nonselfadjoint ~, 325 nonsymmetric ~, 385, 404 orthogonal~,340,375,428,429

positive definite ~, 197, 206 regularized ~, 68 sample covariance ~, 127 sparse symmetric ~, 23 symmetric ~, 4, 26, 34, 36, 40,

54,77,80,87,93,119,171, 194,209,269,276,361,403, 423

unitary ~, 308 Measure

Lebesque ~, 234, 326, 367 Method

double F ~, 127 extending random matrices ~, Fourier ~, 298 inverse Fourier ~, 298 Martingale differences ~, 6, 7,

66, 67, 127, 158, 193, 361, 389

moment equation ~, 480

496 Index

Monte Carlo -, 113, 168 of thinning matrices -,127,405,454 perpendiculars -, 391

N

random determinants -,167,169 random simulations-, 169 stochastic power -, 184 successive approximations -, 74,

94, 155, 163, 166, 166, 182, 214,222,278,361,363,364

REFORM -, 27, 35, 127, 277, 280, 289, 344, 406, 443

regularization ofresolvents -,68, 361

quasi-inversion -, 230 Walked field -, 168

Norm, 403 spectral-, 403, 409, 455, 461

o Observation

independent -, 185 of special structure -, 203

Operator, 230 p

Parallelepiped, 243, 304 Permanent, 168, 169, 180 Perturbations,

matrix -,35 Point

boundary -, 274 Principle

R

invariance -, 27, 34, 128, 168, 180, 291, 296, 344, 394

Rate of convergence, 127 Regularization

parameter -, 68, 159 Representation

martingale -, 325 Shur's -, 263 spectral -, 394, 472

Resolvent -, 25, 27, 53, 205, 345, 361

Roots Cardano -, 48 polynomial -, 48

S Selfaveraging, 65, 325 Series

Taylor-, 53, 226, 361 Sequence of Series -, Set

Borel -, 234, 326, 367 Simulation

Monte Carlo -, 131, 152 Solution, 3 Space

Euclidean -, 229, 243, 304, Hilbert-, 230 Probability -, 2

Symbol Kronecker -,34,99,174,407

Spectrum, 230 Statistics

order -, 183 T Theorem

central limit -, 42, 402 Helly-, 8, 16, 418 limit -, 60, 80 Schur -,38 strong limit -,

Theory operator spectral -, 230 spectral -, 230 V -,118

Transform Fourier -,83,206,314,436,438 Laplace -,52,62, 71, 76,86,87,

91, 94, 95, 154, 162, 360, 362, 364

modified V -, 325, 344, 378, 388

regularized V -, 233, 315, 349, 354, 394, 398

regularized VI -, 318 regularized V2 -, 318 regularized V3 -, 312, 314, 316 regularized V4 -, 316 Stieltjes -, 1, 2, 5, 8, 14, 15,

198, 24, 26, 37-39, 41-43 Stieltjes inverse -, 38, 222

truncated conditional V1 -, 264 truncated conditional V2 -, 264 V -, 232, 233, 240, 252, 260,

264, 320, 388, 391, 392 Vi -,314

V Value

singular -, 216 Variance

empirical generalized -, 5 generalized -, 5

Index 497

REFERENCES

Although most of these references were not consulted in the writing of this book, they are listed here to give the reader an idea of the scope of the theory of random matrices and its applications.

Aizenman, M., Lebowitz, J. L. and Ruelle, D. [ALR] Some Rigorous Results on the Sherington-Kirkpatrick Spin Glass

Model, Commun. Math. Phys. 112,3-20 (1987).

Aivazian, S. A., Buchstaber, V. M., Eniukov, I. S., Meshalkin, L. D. [ABEM] Applied Statistics. Classification and Reduction of Dimensionality

(in Russian) (1989).

Anderson, T. W. [And1] Asymptotic theory for principal component analysis, Ann. Math.

Statist. 34, 122-148 (1963). [And2] An Introduction to Multivariate Statistical Analysis, 2nd ed., Wiley

and Sons, New York (1984).

Arharov, L. V. [Arh] Limit theorems for the characteristic roots of a sample covariance

matrix, Soviet Math. Dokl, 12, 1206-1209 (1971).

Arnold, L. [Arn1] On the asymptotic distribution of the eigenvalues of random ma­

trices, J. Math. Anal. Appl. 20, 262-268 (1960). [Arn2] On Wigner's semicircle law for the eigenvalues of random matrices,

z. Wahr. Verw. Geb. 34, 191-198 (1971).

Bahr, Von and Esseen, C.G [BE] Inequalities for the r-th absolute moment of a sum of random vari­

ables, 1 ::; r ::; 2, Ann. Math. Statist. 36, 1, 299-303 (1965).

Bai, Z. D. [Bail] Convergence rate of expected spectral distributions of large random

matrices. I. Wigner Matrices, Ann. Probab. 21,2,615-648 (1993). [Bai2] Convergence rate of expected spectral distributions of large random

matrices. II. Sample Covariance matrices, Ann. Probab. 21, 2, 649-672 (1993).

434 References

Bai, Z. D. and Yin, Y. Q. [BY] Limiting behavior of the norm products of random matrices and

two problems of German-Hwang, Pmbab. Theory, Related Fields, 73, 555-569 (1986).

Bai, Z. D., Silverstein, J. W. and Yin, Y. Q. [BSY] A note on the largest eigenvalue of a large-dimensional sample co­

variance matrix, J. Multivar. Anal. 26, 166-168 (1998).

Bai, Z. D. and Wu Y. [BWu1] Limiting behavior of M -estimators ofregression coefficients in high

dimensional linear models. I. Scale-dependent Case, Journal of Multivariate Analysis 51,211-239 (1994).

[BWu2] Limiting behavior of M-estimators of regression coefficients in high dimensional linear models. II. Scale-invariant Case, Journal of Mul­tivariate Analysis 51, 240-251 (1994).

Barsov, D. A. [Bar] Limit theorems for traces of sample covariance matrices, Theory

Pmbab. Appl.. 29, 2, 383-391 (1984).

Basor, E. L. [Bas] Distribution functions for random variables for ensembles of posi­

tive hermitian matrices, Communications in Mathematical Physics 188,327-350 (1997).

Bateman, H. and Erdelyi, A. [Bat] Higher Transcendental Functions, vols. 1-3. McGraw Hill, New

York, (1953-1955).

Bellman, R. [Bel] Intmduction to Matrix Analysis, McGraw Hill, New York, (1960).

Bennet, G., Goodman, V. and Newman, C. M. [BGN] Norm of random matrices, Pacific J. Math. 59,359-365 (1975).

Berezin, F. A. [Ber] Some remarks on Wigner distribution, Theoret. Math. Phys. 17,

3, 1163-1175 (1973).

Bergstrom, H. [Berg] On arrangements which determine limit theorems for eigenvalues

of a sample covariance matrix, Multivariate analysis Y(Pmc. Fifth Internat. Bympos., Univ. Pittsburgh. 1978,201-212 North-Holland (1980).

Bessis, D., Itzykson, C. and Zuber, J. [BIZ] Quantum field theory techniques in graphical enumeration, Adv.

Appl. Math. 1, 109-157 (1980).

Bharucha-Reid, A. T. [Bha] Pmbabilistic Methods in Applied Mathematics. Academic Press,

(1970).

References 435

Bharucha-Reid, A. T. and Sambandham M. [BhaS] Random Polynomials. Academic Press, (1986).

Bogachev, L., Molchanov, L. and Pastur L. [BMP] On the density of states of random band matrices, Mat. Zametki,

50,31-42 (1991).

Bogomolny, E., Bohigas, O. and Leboeuf, P. [BBL1] Distribution of roots of random polynomials, Phys. Rev. Lett., 68,

2726-2729 (1992).

Bogomolny, E., Bohigas, O. and Leboeuf, P. [BBL2] Quantum chaotic dynamics and random polynomials, Journal of

Statistical Physics, 85, Nos. 5/6,639-679 (1996).

Boutet de Monvel, A., Khorunzhy, A. and Vasilchuk, V. [BKV] Limiting eigenvalue distribution of random matrices with correlated

entries, Markov Processes Relat. Fields 2, 607-636, (1996).

Boutet de Monvel, A., Pastur, L. and Shcerbina, M. [BPS] On the statistical mechanics approach in the random matrix theory:

Integrated density of states, J. Stat. Phys. 79,3-4,585-613 (1995).

Brezin, E., Itzykson, C., Parisi, G., and Zuber, J. [BIPZ] Planar diagrams, Commun. Math. Phys. 59, 35-51 (1978).

Brezin, E. and Zee, A. [BrZ] Universality of the correlations between eigenvalues of large random

matrices, Nucl. Phys. B. 402, 613-627 (1993).

Broder, A. [Bro] How hard is it to marry at random( on the approximation of the per­

manent), 19th Proceedings of the ACM Symposium on the Theory of Computing, 50-58 (1986).

Brody, T. A., Flores, J., French, J., Mello, P. A., Pandey, A., Wong, S. [BFF] Random-matrix physics: Spectrum and strength fluctuation, Rev.

Modern Phys. 58, 3, 385-479 (1981).

Bronk, B. V. [Brl] Accuracy of the semicircle approximation for the density of eigen­

values of random matrices, J. Math. Phys. 5, 215-220 (1964). [Br2] Exponential ensemble for random matrices, J. Math. Phys. 6,

228-237 (1965).

Brown, B. M. and Eagleson, G. K. [BrE] Martingale convergence to infinitely divisible laws with finite vari­

ances, Trans. Amer. Math. Soc. 162, 2, 449-453 (1971).

Burkholder, D. L. [Bur] Distribution function inequalities for martingales, Trans. Amer.

Math. Soc. 1, 1, 19-42 (1973).

436 References

Camarda H. S. and Georgopulos P. P. [CG] Statistical behaviour of atomic energy levels: agreement with ran­

dom matrix theory, Phys. Rev. Letters 50, 492~495 (1983).

Carrol, J., Ruppert, D. and Stefanski, L. A. [CRS] Measurement Error in Nonlinear Models. Chapman and Hall, NY,

305pp. (1995).

Casati G., Molinari, G. L. and Izrailev, F. M. [CMI] Scaling properties of band random matrices, Phys. Rev. Lett. 64,

1851~1854 (1980).

Chalker, J.T. and Mehlig, B. [ChM] Eigenvector statistics in non-Hermitian random matrix ensembles,

Probability Theory and Related Fields 93, 249~272 (1992).

Chan, T. [Ch] The Wigner semi~circle law and eigenvalues of matrix~valued dif­

fusions, Theoretical Physics 81, 16, 3367~3370 (1998).

Clarkson, K. L. [CIa] Las Vegas algorithms for linear and integer programming when

the dimension is small. J. Assoc. Comput. Mach. 42, 2, 488~ 499(1995).

Cohen, J.E. and Newman, C.M. [CoN] The stability of large random matrices and their products, Ann.

Probab. 12, 283~310 (1984).

Constantinescu, F., Felder, G. M, Gawedzki, K. and Kupiainen, A. [CFGK] Analyticity of density of states, J. Stat. Phys. 48, 3, 365~391

(1987).

Crisanti, A., Paladin, G. and Vulpiani A. [Cra] Products of Random Matrices in Statistical Physics, Springer­

Verlag 88, 113~131 (1993).

Crisanti, A. and Sompolinsky, H. [CS] Dynamics of spin systems with randomly asymmetric bonds:

Langevin dynamics and a spherical model, Phys. Rev. A. 36, 10, 4922~4939 (1987).

Curtiss, J. H. [Curt] Monte Carlo Method, National Bureau of Standards, Applied Math­

ematics Series 12 (1951).

Dagum, G. B. P., Mihail, M., Luby, M. and Vazirani, U. V. [DMLV] Poly tops, permanents, and graphs with large factors, Proceedings

of the 29th Annual IEEE Symposium on Foundations of Computer Science, 412~422 (1982).

References 437

Dauxois, J., Pousse, A. and Romain, Y. [DPR] Asymptotic theory for the principal component analysis of a vec­

tor random function: Some applications to statistical inference, 1. Multivar. Anal. 12, 136-154 (1982).

Deev, A. D. [Dee] Representation of Discriminant Analysis Statistics and asymptotic

expansions when space dimensions are comparable with sample size. Soviet Math. Dokl. 951, 759-762 (1970).

Deift, P., Kriecherbauer, T., Mclaughlin, K. T-R, Venakides, Zhou, X. [DKM1] Uniform Asymptotic for Polynomials Orthogonal with Respect to

Varying Exponential Weights and Applications to Universality Questions in Random Matrix Theory, Comunications on Pure and Applied Mathematics, Vol. LII, 1335-1425 (1999).

[DKM2] Strong Asymptotics of Orthogonal Polynomials with Respect to Exponential Weights, Comunications on Pure and Applied Mathe­matics, Vol. LII, 1491-1552 (1999).

Delyon, F., Simon, B. and Souillard, B. [DSS] From power pure point to continuous spectrum in disordered sys­

tems, Ann. Inst. H. Poincare, 42, 283-309 (1985).

Demetrefi, K. [Dem] Two-dimensional quantum gravity, matrix models and string the­

ory, Int. 1. Mod. Phys. A. 8,1185-1244 (1993).

Derrida, B. [Der] Random Energy Model; An exactly solvable model of disordered

systems, Phys. Rev. A. 24, 2613-2626 (1981).

Dochan, P. [Doch] Mixing Properties and Examples. Springer Verlag (1994).

Donoghue, W. [Don] Monotone Matrix Functions and Analytic Continuation. Springer

Verlag (1974).

Dyson, F. J. [Dys1] The dynamics of disordered linear chain, Phys. Rev. 92, 6, 1331-

1338 (1953). [Dys2] A Brownian-motion model for the eigenvalue of a random matrix,

1. Math. Phys. 3,6,31-60 (1962). [Dys3] Correlations between eigenvalues of a random matrix, Commun.

Math. Phys. 19,235-250 (1970).

Edelman, A. [Edl] Eigenvalues and condition numbers of random matrices, SIAM 1.

Matrix Anal. Appl., 543-560 (1988). [Ed2] The distribution and moments of the smallest eigenvalue of a ran­

dom matrix of Wishart type, Lin. Alg. Appl. 159, 55-80 (1991).

438 References

[Ed3] On the distribution of a scaled condition number, Math. Comput. 58, 185-190 (1992).

[Ed4] The probability that a random real gaussian matrix has k real eigenvalues, related distribution and the circular law, Journal of Multivariate Analysis, 60, 203-232 (1997).

Edelman, A. and Kostlan, E. [EdK] How many roots of a random polynomial are real?, J. Amer. Math.

Soc. 32, 1-37 (1995).

Edelman, A., Kostlan, E. and Shub, M. [EKS] How many eigenvalues of a random matrix are real?, J. Amer.

Math. Soc. 7, I, 247-267 (1994).

Eltinge, J. L. [EL] Errors in variables estimation in the presence of serially correlated

observations, Contemporary Mathematics 112, 73-95 (1990).

Erdos, P. and Spencer, J. [ErS] The Probabilistic Method in Combinatorics. Academic Press, San

Diego, (1974).

Erdos, P. and Turan, P. [ErT] On the distribution of roots of polynomials. Ann. of Math. 51,

105-119 (1950).

Evangelou, S. N. and Katsanos, D. E. [EvK] Energy level statistics in disordered metals with an Anderson tran­

sition, Journal of Statistical Physics, 85, Nos. 5/6, 525-550 (1996).

Faddeev, D. K. and Faddeeva, V. N. [FaF] Computational Methods in Linear Algebra. W.H. Freeman, San

Francisco (1963).

Farrel, R. H. [Far] Techniques of Multivariate Calculation. Lecture Notes on Mathe­

matics. Vol. 520, Springer Verlag(1976).

Feinberg J. and Zee, A. [FZ] Non-gaussian non-hermit an random matrix theory: Phase transi­

tion and addition formalism analysis, Nuclear Physics [FSj. B 501, NO 3, 643-669 (1997).

Fernandez, R., Frolich, J. and Sokal, A. [FFS] Random Walks, Critical Phenomena and Triviality in the Quantum

Field Theory. Springer Verlag, Heidelberg (1992).

Fine J. and Romain, Y. [FR] Reduced principal-component analysis, Math. Operationsforsch. u.

Statist. Ser. statist. 15,493-512 (1984).

Forrester P. J. and Zuk J. A. [FG] Applications of the Dotsenko-Fateev integral in random-matrix

models, Nuclear Physics B 473, 616-630 (1966).

References 439

Fox, D. and Kahn P. B. [FK] Identity of the n-th order spacing distributions for a class of Hamil­

tonian unitary ensembles, Phys, Rev. 134, B1151-B1192 (1964).

Franchesco, P. D., Ginsparg, P. and Zinn-Justin, J. [FGZ] 2D gravity and random matrices. Phys. Rep. 254, (1) (1995).

Fuller W. A. [FUL] Prediction of true values for the measurement error model, Con­

temporary Mathematics 112,41-57 (1990).

Furedi, Z. and Komlos, J. [FK] The eigenvalues of random symmetric matrices, Combinatorica. 1,

2, 233-241 (1981).

Fyodorov, Ya. V. and Alexander D. M. [FA] Scaling properties of localization in random band matrices:

A s-model approach, Phys. Rev. Lett. 67, 2405-2409 (1991).

Fyodorov, Ya. V. and Sommers, H. J. [FS] Statistics of resonance poles, phase shifts and time delays in quan­

tum chaotic scattering: Random matrix approach for systems with broken time-reversal invariance, Journal of Mathematical Physics 38,4, 1918-1981 (1997).

Fyodorov, Ya. V., Khoruzenko, B. and Sommers, H. J. [FKS] Almost-Hermitian random matrices: eigenvalue density in the com­

plex plane, Physics Letters A 226, 1-2,46-51 (1997).

Gaenssler, P., Strobel, J. and Stute, W. [GSS] On central limit theorems for martingale triangular arrays, Acta

Math. Acad. Sci. 3-4, 205-216 (1978).

Gantmacher, F. R. [Gan] The Theory of Matrices, vol. I, II. Chelsea, New York (1959).

Geman, S. A. [Gel] Limit theorem for the norm of random matrices, Ann. Probab. 8,

2,252-261 (1980). [Ge2] The spectral radius of large random matrices, Ann. Probab. 14,

1318-1328 (1986).

Geman, S. A. and Hwang, C. R. [GeH] A chaos hypothesis for some large systems of random equations. Z.

Wahrsch. verw. Gebiete 60, 291-314 (1982).

Ginibre, J. [Gin] Statistical ensembles of complex, quaternion, and real matrices, J.

Math. Phys. 6, 440-449 (1965).

Girko, V. L. [Girl] Limit theorems for permanent of random matrix, Theory Probab.

Math. Statist. 3, 29-34 (1971).

440

[Gir2]

[Gir3]

[Gir4]

[Gir5]

[Gir6]

References

Inequalities for a random determinant and a random permanent, Theory Probab. Math. Statist. 4, 42~50 (1971). Limit theorems for a random determinant I, Theory Probab. Math. Statist., 5, 25~31 (1971). Limit theorems for a random determinant II, Theory Probab. Math. Statist. 6, 39~46 (1972). Refinement of some theorems for a random determinant and per­manent, Theory Probab. Math. Statist. 7, 28~32 (1972). Limit theorems for a random determinant. III, Theory Probab. Math. Statist. 8, 25~30 (1973).

[Gir7] Limit theorems for random quadratic forms. I, Theory Probab. Math. Statist. 9, 63~67 (1973).

[Gir8] Limit theorems for solution of system of linear random equations and the eigenvalues and determinant of random matrices, Sov. Math. Dokl. 14, 1508~1511 (1973).

[Gir9] Limit theorems for random quadratic forms. II, Theory Probab. Math. Statist. 10, 47~54 (1974).

[Girl0] Eigenvalues of random matrices I, Theory Probab. Math. Statist. 11, 8~14 (1974).

[Girll] Limit theorems for eigenvalues of random matrices, Sov. Math. Dokl. 15, 2, 636~639 (1974).

[Gir12] Random Matrices. Kiev University, Kiev (1975). (in Russian). [GirI3] Limit theorems of general type for spectral functions of random

matrix, Theory Probab. Appl. 22, 1, 160~163 (1977). [Gir14] The central limit theorem for random determinants, Theory Probab.

Appl. 24, 4, 729~740 (1978). [GirI5] On the central limit theorem for random determinants, Theory

Probab. Math. Statist. 21, 35~39 (1979). [GirI6] Logarithmic law, Dokl. Akad. Nauk Ukr. SSR, Ser. Mat. 4, 35~39

(1979). (in Russian). [Gir17] Distribution of eigenvalues and eigenvectors of Hermitian random

matrices, Ukr. Mat. Zh., 31 (1979). [Gir 18] On the uniqueness of the solution of the canonical spectral equation,

Ukr. Mat. Zh. 32, 6, 546~548 (1980). [GirI9] Theory of Random Determinants. Kiev University, Kiev (1980).

(in Russian). [Gir20] Stochastic Spectral Equation, in: Lecture Notes in Control and In­

formation Sciences, Stochastic Differential Systems, Filtering and Control, Springer Verlag (1980).

[Gir21] On new method of proving some limit theorems for product of in­dependent random matrices, Theory Probab. Math. Statist. 22 (1980).

[Gir22] The stochastic Ljapunov problem, Theory Probab. Math. Statist. 22 (1980).

References 441

[Gir23] The central limit theorem for random determinants, Theory Probab. Appl. 26,3 (1981).

[Gir24] The polar decomposition of a random matrix, Theory Probab. Math. Statist. 23,21-31 (1981).

[Gir25] Limit theorems for determinants of random Jacobi matrices, Theory Probab. Math. Statist. 24 (1982).

[Gir26] Distribution of eigenvalues and eigenvectors of unitary random ma­trices, Theory Probab. Math. Statist. 25, 13-16 (1982).

[Gir27] The V-transformation, Dokl. Akad. Nauk Ukr.SSR, Ser. Mat. 3, 5-6 (1982). (in Russian).

[Gir28] Eigenvalues of random matrices II, Theory Probab. Math. Statist. 27,31-32 (1983).

[Gir29] On the circle law, Theory Probab. Math. Statist. 28, 15-23 (1983). [Gir30] Spectral theory of nonselfadjoint random matrices, Probability The­

ory and Mathematical Statistics, Proceedings of the USSR-Japan Symposium, Tbilisi, 1982, Lectures Notes Mathematics 1021, 153-156 (1983).

[Gir31] Central limit theorem for spectral functions of random Jacobi ma­trices, Theory Probab. Math. Statist. 29 (1984).

[Gir32] Regularized V-transformation, Theory Probab. Appl. 29, 2, p.416-417 (1984).

[Gir33] Circular law, Theory Probab. Appl. 29,4,694-706 (1984). [Gir34] The elliptic law, Theory Probab. Appl. 30,4,640-651 (1985). [Gir35] Spectral theory of random matrices, Russian Math. Surveys 40, I,

77-120 (1985). [Gir36] Distribution of eigenvalues and eigenvectors of orthogonal random

matrices, Ukr. Mat. Zh. 37, 5, (1985). [Gir37] Random Determinants, The Results of Science and Technology,

Theory of Probability and Mathematical Statistics, Theoretical Cy­bernetics, 24 (1986).

[Gir38] G 2-estimates of spectral functions of covariance matrices, Theory Probab. Math. Statist. 35 (1986).

[Gir39] Introduction in general statistical analysis, Theory Probab. Appl. 32,2, 250-265 (1987).

[Gir40] Elliptic law and elements of G-analysis, Probability Theory and Mathematical Statistics, Proceedings of the Fourth Vilnius Confer­ence, 1, VNU Science Press, Vilnius, 489-509 (1987). (in Russian).

[Gir41] Introduction in general statistical analysis, Proceedings of the First World Congress of the Bernoulli Society, vol. 2,327-330 (1987).

[Gir42] Equation for Stieltjes transforms of limit spectral functions of iso­tropic random matrices, Theory Probab. Math. Statist. 36 (1987).

[Gir43] Two estimators of general statistical analysis, Theory Probab. Math. Statist. 37 (1987).

[Gir44] Multidimensional Statistical Analysis. Vyshcha Shkola, Kiev, (1988). (in Russian).

442

[Gir45]

[Gir46]

[Gir47]

[Gir48]

[Gir49]

[Gir50]

[Gir51]

[Gir52]

[Gir53]

[Gir54]

[Gir55]

[Gir56]

[Gir57]

[Gir58]

[Gir59]

[Gir60]

[Gir61]

[Gir62]

[Gir63]

[Gir64]

References

Spectral Theory of Random Matrices. Nauka, Moscow, (1988). (in Russian). Limit normalized spectral functions of self-adjoint random matrices pencil, Ukr. Mat. Zh. 40, 1, (1988). (in Russian). Asymptotic of the distribution of the spectrum of random matrices, Russian Math. Surveys 44, 4, 3-36 (1989). Limit theorems for distributions of eigenvalues of random symmet­ric matrices, Theory Probab. Math. Statist. 41, 23-29 (1989). Limit theorems for sums of distribution of eigenvalues of random symmetric matrices, Ukr. Mat. Zh. 41, 1, 23-30 (1989). (in Russian). G25 estimator of principal components, Theory Probab. Math. Statist. 42, 1-10 (1990). G-estimates of eigenvalues of covariance matrices, Probability The­ory and Mathematical Statistics, Proceedings of the Fifth Vilnius Conference, Vol. 1, Vilnius (1990). (in Russian). Limit theorems for maximal and minimal eigenvalues of symmetric random matrices, Theory Probab. Appl. 35, 4, 677-690 (1990). G-consistent estimates of traces of matrix resolvents, eigenvalues, and eigenvectors, International Congress of Mathematicians, Ky­oto, Abstracts, Short Communications p. 149 (1990). Theory of Random Determinants. Kluwer Academic Publishers (1990). Theory of Empirical Systems of Equations. Lybid, Kiev (1990). (in Russian). Limit theorems for distributions of eigenvalues of random symmet­ric matrices, Theory Probab. Math. Statist. 41 (1990). G-consistent estimators for eigenvalues and eigenvectors of matrix, Factory's Laboratory 57, 3 (1991). G-consistent estimators for eigenvalues and eigenvectors of matrix, Advances in Soviet Mathematics, Dynamical Systems and Statisti­cal Mechanics 3, 15-55 (1991). Asymptotic behavior of eigenvalues of empirical covariance matrices I, Theory Probab. Math. Statist. 44,37-44 (1992). Canonical spectral equation for singular spectral function of ran­dom matrices, Russian Math. Surveys 48,3, 163-180 (1993). Asymptotic behavior of the eigenvalues of empirical covariance ma­trices 2, Theory Probab. Math. Statist. 45, 36-42 (1993). On a class of Spectral Estimators. Theory of Probability and Math. Statist. 47, 15-22 (1993). Asymptotics of eigenvalues of symmetric random matrices, Math. Notices 54,2,769-779 (1994). General equation for the eigenvalues of empirical covariance matri­ces I, Random Oper. Stoch. Eqs. 2, 1, 13-25 (1994).

References 443

[Gir65) General equation for the eigenvalues of empirical covariance matri­ces II, Random Oper. Stoch. Eqs. 2,2, 175-187 (1994).

[Gir66) The circular law: ten years later, Random Oper. Stoch. Eqs. 2, 3, 235-277 (1994).

[Gir67) The canonical equation for the resolvent of a random matrix with asymptotically independent entries I, Random Oper. Stoch. Eqs. 2, 4, 385-406 (1994).

[Gir68) Canonical equation for empirical covariance matrices, Proceedings of the 14th International Conference on Multidivariate Statistical Analysis, Warsaw 225-247 (1995).

[Gir69) Statistical Analysis of Observations of Increasing Dimension. Klu­wer Academic Publishers 286pp. (1995).

[Gir70) The canonical equation for the resolvent of a random matrix with asymptotically independent entries II, Random Oper. Stoch. Eqs. 3, 1,41-63 (1995).

[Gir71) The method ofrandom determinants for estimating the permanent, Random Oper. Stoch. Eqs. 3, 2, 181-193 (1995).

[Gir72) The Elliptic law: ten years later I, Random Oper. Stoch. Eqs. 3, 3,257-302 (1995).

[Gir73) The Elliptic law: ten years later II, Random Oper. Stoch. Eqs. 3, 4,377-398 (1995).

[Gir74) Canonical spectral equation for the eigenvalues of empirical covari­ance matrices, Proceedings of the 6th Vilnius Conference, Vilnius (1995).

[Gir75] Canonical spectral equation for the empirical covariance matrices. Ukrainian Math. Journal, 47, 9,1176-1189 (1995).

[Gir76] Random Matrices, Handbook of Algebra. Elsevier Science B.V., 27-78 (1995)

[Gir77] Spectral Theory of Minimax Estimation. Acta Applicandae Math­ematicae, 43, 59-69, (1996).

[Gir78] Limit theorems for permanent. Theory of Probability and Mathe­matical Statistics 53, 33-42 (1996).

[Gir79] The Canonical Spectral Equation. Theory of Probability and its Applications, 39, 4, 685-691 (1996).

[Gir80] Canonical Equation for the Resolvent of Empirical Covariance Ma­trix Random Operators and Stochastic Equations, 4, 1, 61-76 (1996).

[Gir81] Strong Law for the eigenvalues and eigenvectors of empirical covari­ance matrices. Random Operators and Stochastic Equations, 4, 2, 179-204 (1996).

[Gir82] Strong Law for the eigenvalues of empirical covariance matrices. Multidimensional Statistical Analysis and Theory of Random Ma­trices. Proceedings of the Sixth Eugene Lukacs Symposium Held in Bowling Green State University, Department of Mathematics and

444 References

Statistics, Bowling Green Ohio. VSP Publishing, Utrecht, 61-92 (1996).

[Gir83] Distribution of spacings of random matrices. Part I: The First Spacing Law for Gaussian Hermitian matrices, Random Operators and Stochastic Equations 4, 3, 283-300 (1996).

[Gir84] Theory of Linear Algebraic Equations with Random Coefficients, Allerton Press, Inc. New York, 320pp. (1996).

[Gir85] A matrix equation for the resolvents of random matrices with in­dependent blocks. Theory of Probability and its Applications, 40, 4,635-644 (1996).

[Gir86] Strong Law for singular values and eigenvectors ofrandom matrices, I. Random Operators and Stochastic Equations 5,1,80-104 (1997).

[Gir87] Strong Circular Law. Random Operators and Stochastic Equations 5,2,173--197 (1997).

[Gir88] Ten years of general statistical analysis. (The main G-estimators of general statistical analysis). Relatorio de pesquisa, RP 43/97, Instituto de Matematica Estatistica e Computac;ao Cientifica, Uni­versidade Estadual de Campinas, Brasil, 97pp. (1997).

[Gir89] Simple proof of the strong circular law. Relatorio de pesquisa, RP 46/97, Instituto de Matematica Estatistica e Computac;ao Cien­tifica, Universidade Estadual de Campinas, Brasil, 22pp. (1997).

[Gir90] Convergence rate of the expected spectral functions of symmetric random matrices equals O(n- 1/ 2)). Relatorio de pesquisa, RP 53/97, Instituto de Matematica Estatistica e Computac;ao Ci­entifica, Universidade Estadual de Campinas, Brasil, 53pp. (1997).

[Gir91] Strong Elliptic Law. Random Operators and Stochastic Equations, 5, 3, 269-306 (1997).

[Gir92] The V-density for eigenvalues of nonsymmetric random matrices and rigorous proof of the strong Circular law, Random Operators and Stochastic Equations, 5, 4, 371-406 (1997).

[Gir93] Refinement of the central limit theorem for random determinants, Theory of Probability and Its Applications, 42, 1, 63-73 (1997).

[Gir94] Strong Law for the singular values and eigenvectors of random ma­trices II. Random Operators and Stochastic Equations, 6, N.3, 1998, 291-310.

[Gir95] Convergence rate of the expected spectral functions of symmetric random matrices equals O(n- 1/ 2). Random Operators and Stochas­tic Equations, 6, N.4, 1998, 359-406.

[Gir96] An Introduction to Statistical Analysis of Random Arrays. Pub­lishing VSP, 673pp.(1998).

[Gir97] Strong Law for the singular values and eigenvectors of random ma­trices III. Inequalities for the spectral radius of large random ma­trices Random Operators and Stochastic Equations, 7, N.2, (1999), 179-200.

References 445

[Gir98] The V -relation between Hermitian and non-Hermitian operators and strong law for normalized spectral functions of non-selfadjoint random matrices with independent row vectors. Markov Processes and Related Fields 4, 4, 499-508 (1998).

[Gir99] Logarithmic law:twenty years later Theory of Probability and Mathematical Statistics, (Ukrainian) N.60, (1999), 17-25.

[Gir100] Strong law and canonical equation for normalized spectral functions of non self adjoint random Jacobi matrices Random Operators and Stochastic Equations, 8, N.1, 2000, 87-104 .

[Gir101] Random block matrix density and SS-Law Random Operators and Stochastic Equations, 8, N.2, 2000, 189-194 .

[Gir102] Twenty five years of stochastic canonical equation for normalized spectral functions of random symmetric matrices Random Opera­tors and Stochastic Equations, 8, N.3, 2000, 275-304 .

[Gir103] Stochastic canonical equation for normalized spectral functions of random Gram matrices Random Operators and Stochastic Equa­tions, 8, N.4, 2000, 397-313 .

[Gir104] The Cubic Law, the Invariance Principle, and related topics in the theory of analytic functions of random matrices Random Operators and Stochastic Equations, 9, N.1, 2001, 53-76.

[Gir105] Universality and semicircular laws for random matrices A + U m

xB(u*)m Random Operators and Stochastic Equations, 9, N.3, 2001, 287-306.

[Gir106] Universality and Arcsine laws for random matrices (A + U B) (A + U B)* Random Operators and Stochastic Equations, 9, N.4, 2001, 339-356.

Girko's Circular Law [GCL] Girko's Circular Law: Let). be Eigenvalues of a set of Random

n x n Matrices. Then).1 v'n is uniformly distributed on the Disk. CRC Concise Encyclopedia of Mathematics on CD-ROM, 1996-9 Eric W. Weisstein

Girko, V. L. and Casati, G. [GiC1] Generalized Wigner law for band random matrices, Random Oper.

Stoch. Eqs. 1, 1 (1993). [GiC2] Limit theorems for band random matrices whose entries have boun­

ded variances, Random Oper. Stoch. Eqs. 1,2,181-191 (1993).

Girko, V. L., Casati, G. and Molinari, L. [GCM] Equation for limit spectral functions of band random matrices, Ran­

dom Oper. Stoch. Eqs. 1, 1, 1-6 (1993).

Girko, V. L. and Dias, R [GiD] Numerical and Monte-Carlo verification of the V-Law, Random Op­

erators and Stochastic Equations, 6, 2, 201-212 (1998).

446 References

Girko, V. L., Ezhov, S. N. and Pluyko, V. A. [GEP1] Integral equations for average scattering matrix, Ukr. Fiz. Zh. 33,

10 (1988). (in Russian). [GEP2] The integral equations for the average stochastic scattering matrix,

Random Oper. Stoch. Eqs. 1,2,161-170 (1993).

Girko, V. L. and Gupta, A. K. [GiG] Asymptotic behavior of spectral function of empirical covariance

matrices, Random Oper. Stoch. Eqs. 2, 1, 43-60 (1994).

Girko, V. L., Kirsch, W. and Kutzelnigg. A. [GKK] Necessary and sufficient conditions for the semicircle law, Random

Oper. Stoch. Eqs. 2,2, 195-202 (1994).

Girko, V. L., Kokobinadze, T. S. and Chajka, O. G. [GDG] Distribution of the eigenvalues of Gaussian random matrices, Ukr.

Mat. Zh. 36, 1,9-12 (1984). (in Russian).

Girko, V. L. and Litvin, LN. [GiL] Stochastic Lyapunov problem for a system of stationary linear dif­

ferential equation, Differential Equations 24, 8, 1307-1324 (1988).

Girko, V. L., Olhovsky, V. S. and Chinarov, V. A. [GOG] To the theory of fluctuation of unitary S-matrix for nuclear reac­

tions, Izv. Akad. Nauk SSSR 48, 1, 166-171 (1984). (in Russian).

Girko, V. L. and Preston, N. [GiP] Numerical and Monte Carlo verification of the first spacing law,

Random Operators and Stochastic Equations 4, 4, 303-314 (1996).

Girko, V. L. and Repin, K. [GiR] One-Quarter-of-circumference Law, Theory of Probability and

Mathematical Statistics 50, 67-71 (1995).

Girko, V. L. and Rosen, D. [GiRo] Asymptotics for the normalized spectral function of matrix quadra­

tic forms, Random Oper. Stoch. Eqs. 2,2, 153-162 (1994).

Girko, V. L. and Stepachno, I. V. [GiS1] An estimator of Stieltjes transform of spectral function of singular

eigenvalues, Ukr. Mat. Zh. 4, 464-469 (1990). (in Russian). [GiS2] Asymptotics of Stieltjes transform of random spectral functions,

Theory Probab. Math. Statist. 46 (1993).

Girko, V. L. and Vasilev, V. V. [GiV1] Limit theorems for determinants of random Jacobi matrices, Theory

Probab. Math. Statist. 24 (1982). [GiV2] Limit theorems for the normalized spectral functions of nonselfad­

joint random Jacobian matrices, Theory Probab. Appl. 30, 1 (1985).

References 447

Girshick, M. A. [Girs] On the sampling theory of roots of determinantal equations, Ann.

Math. Statist. 10,203-224 (1939).

Gnedenko, B. V. and Kolmogorov, A. N. [GnK] Limit Distribution for Sums of Independent Random Variables.

Addison-Wesley, New York (1954).

Gold'sheid, Va. and Guvarc'h, Y. [GG] Zariski closure and the dimension of the Gaussian law of the prod­

uct of random matrices. I, Probab. Theory Relat. Fields 105, 5, 109-142 (1996).

Gold'sheid, Va. and Khoruzhenko, B. A. [GK] Distribution of eigenvalues in Non-Hermitian Anderson models,

Phisical Review Letters, 80, 13, 2897-2900 (1998).

Gold'sheid, Va. and Margulis, G. A. [GoM] Lyapunov indices of a product of random matrices, Russian Math.

Surveys 44,5, 11-71 (1989).

Grenander, U. [Gre] Probabilities on Algebraic Structures. Wiley, New York-London

(1963).

Grenander, U. and Szego, G. [GrS] Toeplitz Forms and Their Applications. University of California

Press, Berkeley, California (1958).

Grenander, U. and Silverstain, J. W. [GrSil] Spectral analysis of networks with random topologies SIAM J.

Appl. Math., 32, 2, 499-519 (1977).

Gudmundsson, T., Kenney, C. and Laub, A. J. [GKL] Small-sample statistical estimates for the sensitivity of eigenvalue

problem, SIAM Journal on Matrix Analysis and Applications 18, 4, (1997).

Guhr, T. [Guh] Transitions toward quantum chaos: with super symmetry from Pois­

son to Gauss, Annals of Physics 250, 145-192 (1996).

Guhr, T. and Wettig T. [GW] An Itzykson-Zuber-like integral and diffusion for complex ordinary

and supermatrices, J. Math. Phys. 37, 12, 6395-6413 (1996).

Haake, F. [Haa] Quantum Signatures of Chaos. Springer Verlag, Heidelberg (1991).

Hammersley, J. M. [Ham] The zeros of a random polynomial, Proceedings of the Third Berke­

ley Symposium on Mathematical Statistics and Probability, Univer­sity of California Press, Berkeley-Los Angeles, 2, 89-111 (1956).

448 References

Hatano, N. and Nelson, D.R. [HaNl] Localization transitions in non-Hermitian quantum mechanics,

Phys. Rev. Lett. 77, 570-573 (1996).

Hatano, N. and Nelson, D.R. [HaN2] Vortex pinning and non-Hermitian quantum mechanics, Phys. Rev.

B56, 8651-8673 (1997).

Hasegawa, H. [Has] Density of states in aperiodic solids, Supplement of the Progress of

Theoretical Physics 53 (1973).

Hertz, J. A. [Her] Statistical dynamics of learning, in: Lecture Notes in Physics, (Ed.

Luis Garrido), Statistical Mechanics of Neural Networks, Proceed­ings, Sitges, Barcelona, Spain, Springer-Velag (1990).

Hotelling, H. [Hot] Analysis of a complex of statistical variables into principal compo­

nents, J. Educ. Psychol. 24, 417-441, 498-520 (1933).

Hwang, C. R. [Hwa] A brief survey on the spectral radius and spectral distribution of

large random matrices with i.i.d. entries. In Random Matrices and their Applications. Contemporary Mathematics 50, 145-152 (1986).

Ibragimov, I. A. and Linnik, Yu. V. [IbL] Independent and Stationary Random Variables, Wolters-Noordhoff,

Groningen (1971).

Ibragimov, I. A. and Maslova , N. B. [IbM] On the expected number of real zeros of random polynomials.I.

Ishii, K.

Coefficients wth zero means, Theory of Probab. and Its Applications 16, 229-248 (1971).

[Ish] Localization of Eigenstates and Transport Phenomena in the One­Dimensi- onal Disordered System, Supplement of the Progress of Theoretical Physics, 53 (1973).

Iso, S. and Kavalov, A. [IK] Breakdown of universality in random matrix models, Nuclear Phy­

sics [FSJ. B 501, NO 3, 670-684 (1997).

Itzykson, C. and Zuber, J. B. [ItZ] The planar approximation II, J. Math. Phys. 21, 3, 411-421

(1980).

Izrailev, F. M. [Izr] Simple models of quantum chaos: spectrum and eigenfunctions,

Phys. Report. 196, N 5-6, 299-392 (1990).

References 449

James, A. T. (Jam] The non-central Wishart distribution, Proc. Roy. Soc. London

229A, 1178,364-366 (1955).

Janik, R. A., Nowak, M. A., Papp, G. and Zahed, I. [JNPZ] Non-hermitan random matrix models, Nuclear Physics [FSj. B

501, NO 3, 603-642 (1997).

Jerrum, M. R. and Sinclair, A. J. [JeS] Approximating the permanent, SIAM J. Comput. 18, 1149-1178

(1989).

Johansson, K. [Joh] On fluctuations of eigenvalues of a random Hermitian matrices,

Duke Mathematical Journal. 91, N1, 151-204 (1998).

Jonsson, D. (Jon] Some limit theorems for the eigenvalues of a sample covariance

matrix, J. Multivar. Anal. 12, 1-38 (1982).

Juhasz F. [J u1] On the asymptotic behaviour of the spectra of non-symmetric ran­

dom (0,1) matrices, Discrete Math. 41, 161-165 (1982). [Ju2] On the characteristic values of nonsymmetric block random matri­

ces, J. Theoret. Probab. 3,2, 199-205 (1990). [Ju3] On the eigenvalues of block random matrices, Linear Algebra and

its Applications, 246, 225-231 (1996). (Ju4] On the spectrum of a random graph. In Algebraic and Discrete

Methods in Graph Theory (L. Lovasz and V.T. Sos, Eds.) Proc. ColI. Math. Soc. J. Bolyai 25, 1, 313--316 (1981).

Kac, M. [Kac] On the average number of real roots of random algebraic equation,

Bull. Amer. Math. Soc. 49, 314-320 (1943).

Kaijeser T. [Kai] A note on random systems with complete connection and their

applications to products of random matrices, Contemporary Math­ematics, Random Matrices and Their Applications, 50, 243-254 (1986).

Karabali, D., Sakita, B. [KS] Semiclassical approach for nonrelativistic fermions in low dimen­

sions, International Journal of Modern Physics, A, 6, 28, 5079-5097 (1991).

Karmarkar, N., Karr, R., Lipton, R., Lovasz, L. and Luby, M. [KKLLL] A Monte Carlo algorithm for estimating the permanent, SIAM J.

Comput. 22, 2, 284-293 (1993).

Karp, R. M., Luby, M. and Mardas, N. [KLM] Monte Carlo approximation algorithms for enumeration problems,

J. Algorithms 10, 429-448 (1989).

450 References

Kasin, B. S. [Kas] On the mean value of certain function connected with the conver­

gence of orthogonal series, Anal. Math. 4, 27-35 (1978).

Kato, T. [Kat] Perturbation Theory for Linear Operators. Springer Verlag, Berlin

(1966).

Kellogg, O. D. [Kel] Foundations of the Potential Theory. Dover, New York (1953).

Khorunzhy, A. and Pastur, L. [KP] Limits of infinite interaction radius, dimensionality and the number

of components for random operators with off-diagonal randomness, Commun. Math. Phys. 153, 605-646 (1993).

Khorunzhy, A., Khoruzhenko, B. and Pastur, L. [KKP1] On the liN corrections to the Green functions of random matri­

ces with independent entries, J. of Physics A. Mathematical and General. 28, L31-L35 (1995).

[KKP2] Asymptotic properties of large random matrices with independent entries, J. Math. Phys. 37, 10,5033-5060 (1996).

Khorunzhy, A., Molchanov, S. and Pastur L. [KMP] On the eigenvalue distribution of band random matrices in the limit

of their infinite order, Teor. Mat. Fiz. 90,163-178 (1992).

Khorunzenko B. [Khor] Large-N eigenvalue distribution of randomly perturbed asymetric

matrices, J. Phys. A: Math. Gen. 29, L165-L169 (1996).

Kollo, T. and Neudecker, H. [KoN1] Asymptotics of eigenvalues and unit-length eigenvectors of sample

variance and correlation matrices, J. Multivar. Anal. 47, 283-300 (1993).

Kollo, T. and Neudecker, H. [KoN2] Asymptotics of eigenvalue-normed eigenvectors of sample variance

and correlation matrices, Proceedings of the sixth Lukacs Sympo­sium, 111--128 (1996).

Komlos, J. [KomI] On the determinant of (0,1) matrices, Studia Scient. Math. Hung.

3, 4 (1967). [Kom2] On the determinant of random matrices, Studia Scient. Math.

Hung. 2, 1-2 (1968).

Konishi, S. [Koni] Asymptotic expansions for the distributions of statistics based on

the sample correlation matrix in principal-component analysis, Hi­roshima Math. J. 9, 647-700 (1979).

References 451

Konyagin, S. v. [Kon] On minimum of modulus of random trigonometric polynomials with

coefficients ±1, Mathematical Notes 56,80-101 (1994).

Kotani, S. [Kot] Lyapunov exponents and spectra for one-dimensional random

Schrodinger operators, Comtemporary Mathematics, Random Ma­trices and Their Applications, 50, 277-286 (1986).

Kuz, M., Lewentein, M. and Haake, F. [KLH] Density of eigenvalues of random band matrices, Phys. Rev. A 44,

5,2800-2808 (1991).

Landkof, N. S. [Lan] Foundations of Modem Potential Theory, Springer-Verlag, 425p.

(1972).

Larsen, F. [Lar] Brown measures and R-diagonal elements in finite von Neumann

algebras, Ph.D. thesis, University of Southern Denmark (1999).

Lehmann, N. and Sommers, H. J. [LeS] Eigenvalue statistics of random real matrices, Phys. Rev. Lett. 67,

8,941-944 (1991).

Lifshits, I. M. [Lif] On the structure of the energy spectrum and quantum state of

disordered condenser systems, Sov. Fiz. Usp. 7, 549-573 (1965). (in Russian).

Lovasz, L. [Lov] On determinants, matchings, and random algorithms, Budach, L.

(editor), Fundamentals of Computing Theory, Akademica Verlag, Berlin (1979).

Lovasz, L. and Plummer, M. D. [LoP] Matching Theory. Academic Press, New York, 1986.

Lubinsky, D. S. [Lub] Strong asymptotics for extremal errors and polynomials associated

with Erdos-type weights. Longman Group UK Limited, 240p., 1989.

Luby, M., Sinclair, A., and Zuckerman, D. [LSZ] Optimal speedup of Las Vegas algorithms, Inf. Process. Lett. 47,

173-180 (1993).

Marcus, M. and Mine, H. [MaM] A Survey of Matrix Theory and Matrix Inequalities. Allyn and

Bacon (1964).

Marchenko, V. A. and Pastur, L. A. [MaP] Distribution of the eigenvalues in certain sets of random matrices,

Mat. Sb. 1,457-483 (1967). (in Russian).

452 References

Mardia, K. V., Kent, J. T. and Bibby, J. M. [MKB] Multivariate Analysis. Academic Press, London (1979).

Marriot, F. M. [MaM] Practical problems in a method of cluster analysis, Biometrics 27,

501-514 (1971).

Megrabian, R. M. [Meg] On a characteristic of random matrices connected with uncondi­

tional convergence almost everywhere, Anal. Math. 14, 37-47 (1988).

Mehta, M. L. [Mehl] Random Matrices and the Statistical Theory of Energy Levels. Aca­

demic Press, New York-London (1967). [Meh2] Random matrices. Academic Press, San Diego (1991).

Memon, A. Z. and Okamoto., M. [MO] . Asymptotic expansion of the distribution of the z statistic in

discriminant analysis. J. Multivariate Analysis 1 294-307 (1971).

Mezineseu G. A., Bessis, D., Fournier, J. D., Mantica, G. and Aaron, F. D.

[MBF] Distribution of Roots of Random Real Generalized Polynomials. Journal of Statistical Physics, 86, 3/4, 59-71 (1997).

Miller, S. M. [Mil] Analysis of residuals from measurement error models, Contempo­

rary Mathematics, 112, 59-71 (1990).

Mine, H. [Min] Permanents. Encyclopedia of Mathematics and its Applications, 6.

Addison-Wesley, Reading (1978).

Mizutori, S. and Zelevinsky, V. G. [MZ] Level and width statistics for a decaying chaotic system, Z. Phys.

A 346, 1-9 (1993).

Molehanov, S. A., Pastur, L. A. and Khorunzhy, A. M. [MPK] Distribution of eigenvalues of random band matrices in the limit of

infinite order, Teor. Mat. Fiz. 90,2 (1992). (in Russian).

Moro, J., Burke, J. V. and Overton, M. L. [MBO] On the Lidskii-Vishik-Lyusternik Perturbation Theory for Eigen­

values of Matrices with Arbitrary Jordan Structure, SIAM Journal on Matrix Analysis and Applications 18, 4, (1997).

Muirhead, R. J. [Muir] Aspects of Multivariate Statistical Theory. Wiley & Sons, New York

(1982).

Muskhelishvili, N. I. [Musl] Singular Integral Equations. Noordhoff, Groningen (1953). [Mus2] Some Principal Problems of Mathematical Theory of Elasticity.

References 453

Nauka, Moscow (1966). (in Russian).

Nelson, David R. and Shnerb, Nadav M. [NS] Non-Hermitian localization and population biology, Phys. Rev. E

(3) 58, 2, part A, 1383-1403 (1998).

Neudecker, H. and Wesselman, A. M. [NW] The asymptotic variance matrix of the sample correlation matrix,

Linear Algebra Appl. 127,589-599 (1990).

Nica, A., and Speicher, R. [NSl] R-diagonal pairs-a common approach to Haar unit aries and circu­

lar elements, Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 149-188 (1997).

[NS2] Commutators of free random variables, Duke Math. J. 92, 3, 553-592 (1998).

Nuquist, H., Rice, S., and Riordan, O. [NRR] The distribution of random determinants, Quart. Appl. Math. 12,

2 (1954).

Okamoto, M. [0] An asymptotic expansion of the distribution of linear discriminant

function. Ann. Math. Stat. 34, 1286-1301 (1963).

Olver, F. W. J. [Olv] Asymptotics and Special Functions. Academic Press, New York

(1974).

Opperman, R. and Wegner, F. [OpW] Disordered system with n orbitals per site: n - 1 expansion, Z.

Phys. 34,327-348 (1979).

Paige, C. C. and Saunders, M. A. [PaS] Solutions of sparse indefinite systems of linear equations, SIAM J.

Numer. Anal. 12, 617-629 (1975).

Papadimitriou, C. H. [Pap] Complexity Theory. Addison-Wesley, Reading, 1994.

Pastur, L. A. [Pasl] Spectra of random self-adjoint operators, Russian Math. Surveys

28, 1,4-63 (1973). [Pas2] Spectral properties of disordered systems in the one-body approxi­

mation, Commun. Math. Phys. 75, 179-196 (1980). [Pas3] On the universality of the level spacing distribution for some en­

semble of random matrices, Lett. Math. Phys. 25, 259-265 (1992).

Pastur, L. A. and Vasilchuk, V. [Pa V] On the law of additional of random matrices, Comm. Math. Phys.

214,2,249-286 (2000).

454 References

Pastur, L. A. and Shcherbina, M. [PaS] Universality of the local eigenvalue statistics for a class of unitary

invariant random matrix ensembles, J. Stat. Phys. 86, 109-147 (1997).

Pastur, L. A. and Figotin, A. [PaF] Spectra of Random and Almost Periodic Operators. Springer Verlag

(1992).

Pastur, L. A. and Khorunzhy, A. M. [PaK] Limit Distribution of Eigenvalues of Random Operators of Wegner

Model, Preprint FTINT 389, Kharkov (1989). (in Russian).

Petrov, v. V. [Pel] Sums of Independent Random Variables. Springer -Verlag, 346pp.

(1975).

Porter, C. E. [Por] Statistical Theories of Spectra: Fluctuations, Collection of Reprints

and Original Papers, Academic Press, New York (1965).

Portnoy S. [Port1] Asymptotic behavior of M-estimators of p regression parameters

when p2 In is large, Ann. Statist. 12, 1298-1309 (1984). [Port2] Asymptotic behavior of M-estimators of p regression parameters

when p2 In is large. II. Normal approximation, Ann. Statist. 13, 1403-1417 (1985).

[Port3] Asymptotic behavior of the empirical distribution of M-estimated residuals from a regression model with many parameters, Ann. Statist. 14, 1152-1170 (1986).

[Port4] On the CLT in RP when p -+ 00, Probab. Theory Related Fields 73, 571-584 (1986).

Poschel, J. [Pos] Examples of discrete Schrodinger operators with pure point spec­

trum, Commun. Math. Phys. 88,447-463 (1983).

Prekopa, A. [Pre] On random determinants I, Studia Scient. Math. Hung. 2, 1-2

(1967).

Preston, N. [Pres] On Girko's Cubic Law, Random Operators and Stochastic Equa­

tions. 6, 1,89-94 (1998).

Reed, M. and Simon, B. [RS] Methods of Modern Mathematical Physics. IV: Analysis of Opera­

tors. Acad. Press. London (1978).

Rice, S. O. [Ric] Mathematical theory of random noise, Bell System Tech. J. 25,

46-156 (1945).

References 455

Ryan, 0. [Ry] On the limit distributions of random matrices with independent or

free entries, Commun. Math. Phys. 193, No 3 595-626(1998).

Schafer L. and Wegner F. [Sch W] Disordered system with n orbitals per site: Lagrange formulation,

Z. Phys. 38, 113-126 (1980).

Schmidt, H. [Sch] Disordered one-dimensional crystals, Phys. Rev. 105, 2, 425-441

(1957).

Selberg, A. A. [Sell Remarks on a multiple integral. (Norwegian) Bemerkninger om et

multippel integral, N orsk M atematisk Tidsskrift 26, 71-78 (1944).

Serdobolsky, V.1. [Serl] On classification errors induced by sampling, Theory of Probability

and its Applications 24, 130-143 (1979). [Ser2] Resolvent and spectral functions of sample covariance matrices of

increasing dimension, Russian Math. Surveys 40, (1985).

Shepp, L. A. and Vanderbei R.J. [SV] The complex zeros of random polynomials, Transactions of the

American Mathematical Society 347, 11, 4365-4384 (1995).

Sherington, D. and Kirkpatrick, S. [SK] Solvable model of a spin glass, Phys. Rev. Lett. 35, 1792-1796

(1975).

Silverstein, J. W. [Sill] The smallest eigenvalue of a large-dimensional Wishart matrix,

Ann. Probab. 13,1364-1368 (1985). [Sil2] On the weak limit of the largest eigenvalue of a large-dimensional

sample covariance matrix, J. Multivar. Anal. 30, 307-311 (1989). [Sil3] Eigenvalues and eigenvectors of large dimensional sample covari­

ance matrices, Contemporary Mathematics, Random Matrices and Their Applications, 50, 153-159 (1986).

Silverstein, J. W. and Bai, Z. D. [SiB] On the empirical distribution of eigenvalues of a class of large

dimensional random matrices, 1. Multivar. Anal. 54, 175-192 (1995).

Silverstein, J. W. and Choi, S. I. [SiC] Analysis of the limiting spectral distribution of large dimensional

matrices, J. Multivar. Anal. 54, 295-309 (1995).

Silverstein, J. W. and Combettes, P. L. [SiCo] Signal detection via spectral theory of large dimensional random

matrices, IEEE Truns. Signal Processing 40, 2100-2105 (1995).

456 References

Sinai, Ya. G. and Soshnikov, A. [SiS] A refinement of Wigner's semicircle law in a neighborhood of the

spectrum edge for random symmetric matrices, Functional Analysis and its Applications 60, 114-131 (1998).

Sinclair, A. J. [Sin] Algorithms for Random Generation and Counting: A Markov

Chain Approach. Birkhauser, Basel (1993).

Sommers, H. J., Crisanti, A., Sompolinsky, H. and Stein, Y. [SCSS] Spectrum of large random asymmetric matrices, Phys. Rev. Lett.

60, 1895-1898 (1988).

Soshnikov, A. [Sosl] Level spacings distribution for large random matrices: Gaussian

fluctuations, Annals of Mathematics 60, 573-617 (1998).

Sparo, D. I. and Sur, M. G.

Stein, C.

[SS] On the distribution of roots of random polynomials, Vestnik Mos­kov. Univ. Ser. I Mat. Mekh. 112, 40-53 (1962).

[Stl] Inadmissibility of the usual estimator for the mean of a multivariate normal distribution, Proc. 3rd Berkeley Symp. Math. Stat. Prob. 1, 197-206 (1955).

[St2] The admissibility of Hotellings T2 test, Ann. Math. Stat. 27, 616-623 (1956).

[St3] Inadmissibility of the usual estimator for the variance of a normal distribution with unknown mean, Ann. Inst. Math. 16, 155-160 (1964).

Stockmann, H. J. and Seba, P. [SS] The joint energy distribution function for the Hamiltonian H =

Ho- C WW+ for the one-channel case, J. Phys. A: Math. Gen. 31,No 15, 3439-3448 (1998).

Stone, A. D., Mello, P. A., Muttalib, K. A. and Pichard, J. L. [SMMP] Random theory and maximum entropy models for disordered con­

ductors. In Mesoscopic Phenomena in Solids. 56, eds. B.L. Altshuler, P.A. Lee, and R.A. Webb, North-Holland, Amsterdam, Ch.9, 369-448 (1991).

Sugiura, N. [Sug] Asymptotic expansions of the distributions of the latent roots and

latent vectors of the Wishart and multivariate F-matrices, J. Mul­tivar. Anal. 6, 500-525 (1976).

Szego, G. [Sze] Orthogonal Polynomials, American Mathematical Society, Provi­

dence, (1939).

References 457

Tikhonov, A. N., Goncharsky, A. V., Stepanov, V. V., Yagola, A. G. [TGSY] Numerical Methods for the Solution of Ill-Posed Problems, Kluwer,

252p. (1995).

Tracy, A. C. and Widom, H. [TrWl] Level spacing distributions and the Airy kernel, Commun. Math.

Phys. 159, 151-174 (1994). [TrW2] Level spacing distributions and the Bessel kernel, Commun. Math.

Phys. 161, 289-309 (1994). [TrW3] Fredholm determinants, differential equations and matrix models,

Commun. Math. Phys. 163,33-72 (1994). [TrW4] On orthogonal and symplectic matrix ensembles, Commun. Math.

Phys. 177, 727-754 (1996).

Tyler, D. [Tyll] Asymptotic inference for eigenvectors, Ann. Statist. 9, 725-736

(1981). [TyI2] Robustness and efficiency properties of scatter matrices, Biometrika

70,411-420 (1983).

Valiant, L. G. [Vall] The complexity of computing the permanent, Theor. Compo Sci.

8, 189-201 (1979). [VaI2] The complexity of enumeration and reliability problems, SIAM 1.

Comput. 8,410-421 (1979).

Vere-Jones, D. rVer] Alpha-permanents and their applications to multivariate Gamma,

negative binomial and ordinary binomial distributions, New Zealand Journal of Mathematics 26,125-149 (1997).

Vilenkin, N. J a. [ViI] Special Functions and Theory of Group Representations, American

Mathematical Society, Providence (1968).

Voiculescu, D. V., Dykema, K. J., and Nica, A. [VDN] Free random variables. American Mathematical Society, Provi­

dence, RI (1992).

Wachter, K. W. [Wac] The strong limits of random matrix spectra for sample matrices of

independent elements, Ann. Probab. 6,1-18 (1978).

Wald, A. [Wall Estimation of a parameter when the number of unknown param­

eters increases indefinitely with the number of observations, Ann. Math. Statist. 19, 2, 220-227 (1948)

Watson, G. N. [Wat] A Treatise on the Theory of Bessel Functions, Cambridge, 1922.

458 References

Wegmann, R. [We] The asymptotic eigenvalue-distribution for a certain class of random

matrices, J. Math. Anal. Appl. 56, 113-132 (1976).

Wegner, F. [Weg] Disordered system with n orbitals per site: n = 00 limit, Phys.

Rev. B. 19, 2, 783-792 (1979)

Whittermore A. S. [WH] Errors-in-variables regression problems in epidemiology, Contemp.

Mathematics 112, 17-31 (1990).

Wigner, E. P. [Wig1] Statistical properties of real symmetric matrices with many dimen­

sions, Can. Math. Pmc., 174-198 (1957). [Wig2] On the distribution of the roots of certain symmetric matrices, Ann.

Math. 67,2,325-327 (1958). [Wig3] Random matrices in physics, SIAM Rev. 9, 1, 1-23 (1967).

Wilkins, J. E. jun. [Wilk] The expected value of the number of real zeros of random sum

of Legendre polynomials. Pmc. Am. Math. 125, 5, 1531-1536 (1997).

Wilkinson, J. H. [Will The Algebraic Eigenvalue Pmblem, Clarendon Press, Oxford

(1965).

Wilkinson, J. H. and Reinsch, G. [WiR] Algebra. Springer Verlag (1971).

Yin, Y. Q. [Yin] Limiting spectral distribution for a class of random matrices, J.

Multivar. Anal. 20, 50-68 (1986).

Yin, Y. Q., Bai, Z. D. [YB] Spectra for large-dimensional random matrices, Contemporary

mathematics, Random Matrices and Their Applications, 50, 161-167 (1986).

Yin, Y. Q., Bai, Z. D. and Krishnaiah, P. R. [YBK] On limit of the largest eigenvalue of the large-dimensional sample

covariance matrix, Tech. Report No. 84-44, Pittsburgh University, Pittsburgh (1984).

Yin, Y. Q. and Krishnaiah, P. R. [YiK1] A limit theorem for the eigenvalues of product of two random ma­

trices, J. Multivar. Anal. 13, 489-507 (1984). [YiK2] Limit theorem for the eigenvalues of the sample covariance matrix

when the underlying distribution is isotropic, Theory Pmbab. Appl. 30,861-867 (1985).

Index

A Algebra

(]" -, 26, 68, 93, 253 Algorithm

Gauss -, 278, 279 Angles

Euler -, 395, 399 B Block

C

random -, 304 symmetric -, 305 independent -, 309

Class of distributions C1 -,393 C2 -, 393 C3 -, 393 C4 -, 393 C5 -, 394 C6 -, 394 C7 -, 394 C8 -, 394 C9 -, 395 ClD -, 395 C11 -, 399, 415

Class of estimators 6 8 -, 308

Coefficient asymptotic independence -, 22,

305, 307 Gofactor, 314 Condition

block asymptotic independence -, 26

of the uniform asymptotic neg­ligibility -, 86, 135

G-, 33, 282, 298, 341, 344, 349

459

Lindeberg -, 2, 35, 46, 48, 285, 299, 304, 357, 399

Ljapunov's -, 61, 394, 421, Continuation

analytic -, 120, 292, 426 Convergence

weak -,270 Coordinates

Cartesian -, 397 Criterion

quality -, 268 D Determinant, 33 253, 342

random-, 59, 60, 342 Distribution

Cauchy -, 212 Haar -,383 finite-dimensional -, 301 limit spectral -, 33 normal -, 273 multidimensional -, 327 partial -, 145

Density

E

block matrix -, 12 BMP -,52 Cauchy -, 212 Cubic -,365 G-probability -, 58

Eigenvalue, 33, 346 extreme -, 372 minimal -, 343

Eigenvector 384, 388 Entries

stationary -, 15 asymptotically constant -, 69,

70, 867

460

Equation C1 -,386 C2 -, 387 C3 -, 390 characteristic -, 33 differential -, 237 L1 -,385 L2-, 387 L3 -, 389 L5 -, 355 spectral stochastic -, 77

Equation canonical K31 -, 3, 6, 8 K32 -, 10, 11, 16, 23 K33 -, 18, 24 K34 -, 27, 32 K35 -, 36, 41 K36 -,47 K37 -, 55, 56 K38 -, 58 K39 -, 71, 85, 88, 101 K 40 -, 105 K41 -, 161 K42 -, 166, 168 K43 -, 172 K 44 -,174 K45 -, 178, 186 K46 -, 188 K47 -, 206, 210, 211 K 48 -,217 K49 -, 220 K50 -, 226, 235 K51 -, 247, 249 K52 -, 262

Index

K53 -, 265, 299, 300, 302, 304 K54 -, 323 K55 -, 345, 350 K56 -, 361, 362 K57 -, 375, 377 K58 -, 407, 409 K59 -, 421, 422, 426 linear algebraic -, 280 linear differential -, 351 stable -, 156 stochastic -, 300

Estimator, 268

consistent -, 278 G8 -, 308, 309, 310 Modified G8 -, 308 G-consistent -, 363

Expectation conditional -, 68, 93, 143, 314,

341 F Form

quadratic random -, 72, 90, 115, 139, 370, 407, 425

linear -, 305, 309 Formula

Cardano -, 381 Cramer -, 271 first auxiliary -, 415 G-, 278, 279 inverse -, 313 matrix perturbation -, 401 perturbation of resolvent -,116 regularized -, 280 second auxiliary -, 417 third class of auxiliary -, 418

Function analytic -, 42, 360, 382 analytic matrix -, 307 characteristic -, 71, 132, 155,

255, 301 Dirac -, 47, 52 inverse -, 360, 385, 388, 390 normalized spectral -, 33, 34,

55,104,219,311,312,339 spectral -, 2 individual -, 13, 137, 157, 359,

360, 413 Functional

G

accompanying -, 122 degenerate -, 100 linear -, 71, 77, 84, 89, 97, 103,

123, 129, 301 nonnegative linear -, 302 stable -, 101, 154

Group of matrices, 397 Group of orthogonal matrices -, 269,

393

GSA,282 I Independence asymptotic,

block~, 5 Inequality

Burkholder ~, 319, 368 Cassini ~, 384 Cauchy~Schwarz~Bunhakowski

~, 287 Gershgorin~, 384

Integral

L Law

Cauchy ~, 351, 370 line ~, 337

Arcsine ~, 214, 415 Arctangent ~, 269, 270 Cauchy ~, 212 Canonical, 270 Circular ~, 378 Cubic ~, 365, 378 first ~, 384 second ~, 343, 386 third ~, 388 G~, 58, 68 independency ~, 382 inverse tangent ~, 270 large numbers ~, 251 logarithmic ~, 59, 60 normal~,64,217,222,228,253,

278

Index

random infinitely divisible~, 105 one quarter ~, 9, 14 pencil ~, 48 positive Cauchy ~, 52 stable ~, 155 standard Normal ~, 315 Strong ~, 216, 221, 238, 242,

313, 340 Strong self-averaging ~, 365 universality ~, 414, 415 Wigner ~, 413

Lemma Borel-Cantelli ~, 319

Limit theorem central ~, 67

461

Logarithm

M

expected ~, 240 regularized ~, 241

Martingale differences 67, 144, 240, 319

Matrix ACE ~, 69, 70, 86, 87, 103 ACE-Gram ~, 103 band~, 203 block ~, 1, 15, 163, 173, 306,

307 complex~, 366 covariance~, 25, 26, 33, 53, 159,

177; diagonal ~, 170, 368 double Gram ~, 365 empirical covariance~, 25, 159,

173 function ~, 306 G ~, 242, 344 Gaussian ~, 369, 380, 384, 410 Gaussian Gram ~, 369 Gaussian nonsymmetric ~, 375 Gram~, 9,15,103,302,365 Hankel ~, 251, 257, 258 Hermitian ~, 1, 275, 276, 366,

382, 393 Hermitian Gaussian ~, 393 Jacobi ~, 187, 211, 311 Jacobi nonsymmetric ~, 311 non Hermitian~, 325 nonselfadjoint ~, 276 nonsymmetric ~, 1, 169 orthogonal ~, 346, 393 pencil ~, 35, 47, 346 positive definite ~, 30 rectangular ~, 390 S~, 225 scattering ~, 225, 394 Szeg6~, 259 symmetric ~, 7, 203,384 Toeplitz ~, 254, 258, 261 tridiagonal ~, 187 unitary ~, 393

Matrizant

462

Hermitian -, 215 Unitary -, 219 Stochastic -, 237, 240, 248

Measure Gaussian -, 393 Haar -, 269, 393 Lebesque -, 269, 314

Method

Index

Hermitization -, 274, 282 least squares -, 267, 268 stochastic least squares -, 268 Martingale differences -, 92 non Hermitian -, 323 normal random regularization -

,60 of thinning matrices -, 26, 164 perpendiculars -, 59 random perturbation -,63,261 successive approximations -, 152 REFORM -,216,242,283,367,

415 regularization of resolvents -, 94,

146, 402 Model

Anderson -, 329 stochastic Leontief -, 274

Moment joint -, 301

N Norm, 268, 391, 404 o Observation

dependent -, 25 independent -, 34, 46, 53, 159,

309 p

Parameter reqularization -, 382

Perturbation formula -, 401

Principle invariance -, 38, 217, 222, 243,

345, 368 Problem Sturm-Liouville -, 197, 199 Progression

geometrical -, 404 Pseudosolution, 280 R Regularization

parameter -, 41 Representation

integral -, 42, 180, 198, 201, 233, 254, 400, 420

spectral -, 346 Roots, 33

square -, 400, 401, 420 S Selfaveraging, 189 Simulation, 278 SLAE, 280, 310 SLAERC, 265, 269, 277, 298, 300,

304 Solution

regularized -,307,309,310,361 Space

Euclidean -, 397 probability -, 26, 242, 314, 340,

344 Spin, 225 Symbol

Kronecker -, 62, 252 T Theorem

limit -, 329, 338 Sturm -,205 Szego -, 258

Theorem V -, 342

Transform first Victory -, 274 Fourier -, 19, 20, 172 Laplace -, 105, 138, 171, 179,

186, 302, 363 modified V -, 238 inverse V -, 340 regularized V -, 344 regularized VI -, 332 Stieltjes -, 2, 4, 10, 11, 23, 24,

27, 28, 32, 34, 36, 45, 46, 49, 71, 88, 100, 101, 105,

Index

122,138,155,159,178,179, 312

reqularized -, 36 Stieltjes inverse -, 51 Victory -, 282, 312, 324, 415 VI -, 275, 283 V2 -, 283 V3 -, 283

Transform of the system VI -,352

v

V2 -, 353 V2-regularized -, 354 V3 -, 353

V-transform, 59 regularized -, 241 triply regularized -, 335

Value principal -, 278, 401 singular -, 355

Variable infinitely divisible -, 87, 142 infinitesimal -, 104, 137, 256 output -, 266 pseudorandom -, 278

Vector input -, 266, 267

463