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Optimization
w. Hildenbrand
I.A. lbragimov Y.A. Rozanov
Springer-Verlag New York Heidelberg Berlin
lA. Ibragimov Lomi Fontanka 25 Leningrad 191011 U.S.S.R.
Editorial Board
A. V. Balakrishnan Systems Science Department University of Calif
omia Los Angeles, California 90024 USA
Y.A. Rozanov V.A. Steklov Mathematics Institute Zazilov St. 42
Moscow 3-333 U.S.S.R.
W. Hildenbrand Institut fur Gesellschafts und
Wirtschaftswissenschaften der Universitat Bonn 0-5300 Bonn
Adenauerallee 24-26 German Federal Republic
AMS Subject Classifications: 6OGlO, 6OGl5, 6OG35
Library of Congress Cataloging in Publication Data
Ibragimov, II'dar Abdulovich. Gaussian random processes.
(Applications of mathematics; 9) Translation of Gaussovskie
sluchainye protsessy. Bibliography: p. 1. Stochastic
processes.
Antol'evich, joint author. QA274.4.I2613 519.2
I. Rozanov, Iurii II. Title.
78-6705
The original Russian edition GAUSSOVSKIE SLUCHAINYE PROTSESSY was
published in 1970 by Nauka.
All rights reserved.
No part of this book may be translated or reproduced in any form
without written permission from Springer-Verlag.
© 1978 by Springer-Verlag New York Inc. Softcover reprint of the
hardcover 1st edition 1978
9 8 7 6 5 4 3 2 I
ISBN-13: 978-1-4612-6277-0 e-ISBN-13: 978-1-4612-6275-6 001:
10.1007/978-1-4612-6275-6
To Andrei Nickolajevich Kolmogorov
Preface
The book deals mainly with three problems involving Gaussian
stationary processes. The first problem consists of clarifying the
conditions for mutual absolute continuity (equivalence) of
probability distributions of a "random process segment" and of
finding effective formulas for densities of the equiva lent
distributions. Our second problem is to describe the classes of
spectral measures corresponding in some sense to regular stationary
processes (in par ticular, satisfying the well-known "strong
mixing condition") as well as to describe the subclasses associated
with "mixing rate". The third problem involves estimation of an
unknown mean value of a random process, this random process being
stationary except for its mean, i.e., it is the problem of
"distinguishing a signal from stationary noise". Furthermore, we
give here auxiliary information (on distributions in Hilbert
spaces, properties of sam ple functions, theorems on functions of
a complex variable, etc.).
Since 1958 many mathematicians have studied the problem of
equivalence of various infinite-dimensional Gaussian distributions
(detailed and sys tematic presentation of the basic results can be
found, for instance, in [23]). In this book we have considered
Gaussian stationary processes and arrived, we believe, at rather
definite solutions.
The second problem mentioned above is closely related with problems
involving ergodic theory of Gaussian dynamic systems as well as
prediction theory of stationary processes. From a probabilistic
point of view, this prob lem involves the conditions for weak
dependence of the "future" of the proc ess on its "past". The
employment of these conditions has resulted in a fruit ful theory
of limit theorems for weakly dependent variables (see, for
instance, [14], [22]); the best known condition of this kind is
obviously the so-called condition of "strong mixing". The problems
arising in considering regularity conditions reduce in the case of
Gaussian processes to a peculiar approxima-
vii
Preface
tion problem related to linear spectral theory. The book contains
the results of investigations of this problem which helped solve it
almost completely.
The problem of estimating the mean is perhaps the oldest and most
widely known in mathematical statistics. There are two approaches
to the solution of this problem: first, the best unbiased estimates
can be constructed on the basis of the spectral density of
stationary noise; otherwise the least squares method can be
applied.
We suggest one common class of "pseudobest" estimates to include
best unbiased estimates as well as classical least squares
estimates. For these "pseudobest" estimates explicit expressions
are given, consistency conditions are found, asymptotic formulas
are derived for the error correlation matrix, and conditions for
asymptotic effectiveness are defined. It should be men tioned that
the results relevant to regularity conditions and the mean estima
tion are formulated in spectral terms and can automatically be
carried over (within the "linear theory") to arbitrary wide-sense
stationary processes.
Each chapter has its own numbering of formulas, theorems, etc. For
ex ample, formula (4.21) means formula 21 of Section 4 of the same
chapter where the reference is made. For the convenience of the
reader we provide references to textbooks or reference books. The
references are listed at the end of the book.
viii
Contents
CHAPTER I Preliminaries 1
1.1 Gaussian Probability Distribution in a Euclidean Space 1.2
Gaussian Random Functions with Prescribed Probability M~~ 2
1.3 Lemmas on the Convergence of Gaussian Variables 5 1.4 Gaussian
Variables in a Hilbert Space 7 1.5 Conditional Probability
Distributions and Conditional
Expectations 13 1.6 Gaussian Stationary Processes and the Spectral
Representation 16
CHAPTER II The Structures of the Spaces H(T) and LT(F)
11.1 Preliminaries 11.2 The Spaces L +(F) and L -(F) 11.3 The
Construction of Spaces Lr(F) When T Is a Finite
Interval 11.4 The Projection of L +(F) on L -(F) 11.5 The Structure
of the a-algebra of Events U( T)
CHAPTER III
Equivalent Gaussian Distributions and their Densities 63
111.1 Preliminaries 63 111.2 Some Conditions for Gaussian Measures
to be Equivalent 74 111.3 General Conditions for Equivalence and
Formulas for
Density of Equivalent Distributions 85 111.4 Further Investigation
of Equivalence Conditions 90
ix
Contents
IV.3 Conditions for Information Regularity IV.4 Conditions for
Absolute Regularity and Processes with
Discrete Time IV.5 Conditions for Absolute Regularity and Processes
with
Continuous Time
CHAPTER V
Complete Regularity and Processes with Discrete Time 144
V.1 Definitions and Preliminary Constructions with Examples 144 V.2
The First Method of Study: Helson-Sarason's Theorem 147 V.3 The
Second Method of Study: Local Conditions 153 V.4 Local Conditions
(continued) 163 V.5 Corollaries to the Basic Theorems with Examples
177 V.6 Intensive Mixing 181
CHAPTER VI
Complete Regularity and Processes with Continuous Time 191
VI.1 Introduction 191 VI.2 The Investigation of a Particular
Function y(T; /1) 195 VI.3 The Proof of the Basic Theorem on
Necessity 200 VI.4 The Behavior of the Spectral Density on the
Entire Line 207 VI.5 Sufficiency 212 VI.6 A Special Class of
Stationary Processes 217
CHAPTER VII
Filtering and Estimation of the Mean
VII.1 Unbiased Estimates VII.2 Estimation of the Mean Value and the
Method of
Least Squares VII.3 Consistent Pseudo-Best Estimates VII.4
Estimation of Regression Coefficients
References
x
224
224
1.1 Gaussian Probability Distributions in a Euclidean Space
A probability distribution P in an n-dimensional vector space [Rn
is said to be Gaussian if the characteristic function
<p(u) = flR n ei(u, X)P(dx), U E ~n
(here (u, x) = LUkXk denotes the scalar product of vectors u = (Ub
... , un) and x = (Xb ... , xn)) has the form
U E [Rn, (1.1)
where a = (ab ... , an) E [Rn is the mean and B is a linear
self-adjoint non negative definite operator called a correlation
operator; the matrix {Bd de fining B is said to be a correlation
matrix. In this case
(U, a) = r (u, x)P(dx), JlRn
(Bu, v) = flR n [(u, x) - (u, a)] [(v, x) - (v, a)]P(dx),
(1.2)
u, V E [Rn.
The distribution P with mean value a and correlation operator B is
con centrated in an m-dimensional hyperplane L of [Rn (m being the
rank of the correlation matrix), which can be expressed as
L = a + B[Rn
(L being the totality of all vectors Y E [Rn of the form y = a +
Bx, x E [Rn).
1
I Preliminaries
In fact,
the distribution P being absolutely continuous with respect to
Lebesgue measure dy in the hyperplane L, so that
P(F) = r p(y)dy, JrnL
(1.3)
where the distribution density p(y), y E L, has the form
1 { 1 -1 } P(Y)=(2nrI2detBexp -"2(B (y-a),(y-a)). (1.4)
Here det B denotes the determinant of the matrix that prescribes
the operator B in the subspace [Rm = B[Rn, and B- 1 is the inverse
on this subspace.
1.2 Gaussian Random Functions with Prescribed Probability
Measure
Let (Q, W, P) be a probability space, i.e., a measurable space of
elements W E Q with probability measure P on a a-algebra W of sets
A £; Q.
Any real measurable function ~ = ~(w) on a space Q is said to be a
random variable. The totality of random variables ~(t) = ~(w, t)
(parameter t runs through a set T) is said to be a random function
of parameter t E T. The random variables ~(t) themselves are said
to be values of this random func tion ~ = ~(t); for fixed WE Q the
real function ~(w, .) = ~(w, t) of t E T is said to be a sample
function or a trajectory of the random function ~ = ~(t).
We shall consider another space X of real functions x = x(t) of t E
T, which includes all trajectories ~ = ~(w, t), t E T, of the
random function ~ = ~(t). (For instance, the space X = [RT of all
real functions x = x(t), t E T, possesses this property.) Denote by
!S the minimal a-algebra of sets of X containing all cylinder sets
of this space, i.e., sets of the form
(2.1)
(the set indicated by (2.1) consists of the functions x = x(t) for
which the values [x(t 1), ... , x(tn)] at the points t b ... , tn E
T prescribe a vector that belongs to a Borel set r in an
n-dimensional vector space [Rn). The mapping ~ = ~(w) under which
each WE Q corresponds to a pertinent sample func tion ~(w, .) =
~(w, t) of t E T -an element of the space X-is a measurable mapping
in a probability space (Q, W, P) onto a measurable space (X, !S).
The sets A E m of the form A = {~ E B}-the preimages of sets B E !S
under the mapping ~ = ~(w, .) indicated-form (in the aggregate) a
a-algebra. This a-algebra m~ is minimal among a-algebras of the
sets containing all sets of the form
(2.2)
2
1.2 Gaussian Random Functions with Prescribed Probability
Measure
(the set indicated consists of the elements W E Q for which the
values [~(w, t1), ..• , ~(w, tn)] prescribe a vector belonging to a
Borel set r of an n-dimensional vector space []~n, or, in other
words, the O"-algebra m:~ is to be generated by values ~(t), t E T.
Probability measure P~ defined on a O"-algebra ~ by the
relation
P~(B) = Pg E B}, BE~, (2.3)
is said to be a probability distribution of the random function ~ =
~(t) (on the pertinent function space X).
We shall discuss next the question: When is the family of real
variables ~(t) = ~(w, t) given on a space Q (parameter t runs
through a set T) a random function with the given probability
distribution P~? More precisely, when does there exist probability
measure P in the space Q related with the given distribution P~ by
means of (2.3)? We assume in this case that the set ~(Q) of all
sample functions ~(w, .) = ~(w, t) of t E T belongs to the space
X.
It is readily seen that such a probability measure P exists if and
only if the set ~(Q) has a complete exterior measure, i.e.,
P~(B) = 1 for B 2 ~(Q) (2.4)
for any measurable set B E X. In fact, if P~ is the probability
distribution of the random function ~ =
~(t), for any set BE X in the complement of the set ~(Q) the set {~
E B} is empty and
P~(B) = Pg E B} = 0.
On the other hand, for any sets Bb B2 E ~,such that {~E Bd = {~E B
2}, the symmetric difference B1 0 B2 = (B1\B2) u (B2\B1) is
contained in the complement of the set ~(Q), and, under the
condition (2.4), P~(B1 0 B2) = 0, P~(B1) = P~(B2)' Hence the
relation
Pg E B} = P~(B), BE~, (2.5)
defines the single-valued function P = peA) on the O"-algebra m:~
of all sets of the form A = {~ E B}, B E ~, generated by ~(t), t E
T. Obviously, P is a probability measure and ~ = ~(t) is a random
function on a probability space (Q, m:, P) with the given
probability distribution P~.
The measure P on the O"-algebra m:~ generated by the variables
~(t), t E T, can be defined uniquely by finite-dimensional
distributions PI! ..... In of which each is a Borel measure on IRn
defined by
(2.6)
PI! ..... In being the probability distribution of the random
vector [~(td, ... , Wn)]. In fact,
peA) = inf I peAk), (2.7) k
3
I Preliminaries
where the lower bound is taken over all sets Ak of the form (2.2),
whose union covers the set A E~. In particular, this fact refers to
the probability distribution P~ on the corresponding function space
X ~it is the probability measure on the a-algebra '8 generated by
the given variables ~(t) = ~(x, t) on the space X, i.e., the
variables of the form
~(t, x) = x(t), XEX (2.8)
(where the parameter t, fixed for each functional ~(x, t) = x(t) of
x E X, runs through the set T).
Denote by r x IRn - m the Borel set in an n-dimensional space of
vectors [x(t!), ... , x(tn)] such that [x(ti,), ... , x(tiJ] E r (r
is a Borel set in an m dimensional subspace IRm s; IRn) with the
remaining coordinates x(tJ arbi trary. Finite-dimensional
distributions are "compatible" in the sense that
ptl •... , tjr X IRn-m) = Ptil "'" tiJn (2.9)
for all sets of the above type. Let X = IRT be the space of all
real functions x = x(t), t E T. According
to a well-known theorem due to Kolmogorov, * any given family of
distri butions p tIo ... , tn prescribes a continuous additive
function P (defined by (2.6), where the variables have the explicit
form (2.8)), on the algebra of all cylinder sets (2.1). This
function extends uniquely to a probability measure P on the
a-algebra '8. The random function ~ = ~(t) with values ~(t) =
~(x, t) in the probability space (X, '8, P) has finite-dimensional
distributions coinciding with the initial compatible distributions
Ptl , ... ,tn'
Starting from probability distribution P = P¢ on the function space
X, under the condition (2.4) we can define (see (2.5)) a
probability measure on the corresponding space Q.
The random functions ~ = W) and ~ = ~(t) with values in the same
space are said to be equivalent if with probability one (for almost
all WE Q)
~(w, t) = ~(W, t)
for each fixed t E T. Obviously, the finite-dimensional
distributions of equiv alent random functions coincide. Taking an
equivalent random function ~ = ~(t) with the trajectories in any
function space X, we can define (see (2.3)) a probability measure
in this space as well.
Random variables are said to be Gaussian if their
finite-dimensional dis tributions are Gaussian. More precisely,
(when we deal with a random function ~ = ~(t) with parameter t E T
under some parametrization), the values ~(t) = ~(w, t) and the
function ~ = ~(t) itself are said to be Gaussian if all
finite-dimensional distributions Ptl , ... , tn are Gaussian.
Probability measure P on a a-algebra ~~ generated by all ~(t) is
also said to be Gaussian.
Each of the finite-dimensional distributions Ptl , ... , tn of the
Gaussian random function ~ = ~(t) has mean value [a(t!), ... ,
a(tn)] and correlation
* See [10], p. 150.
I.3 Lemmas on the Convergence of Gaussian Variables
matrix {B(tk' t j )} where a(t), t E T, is the mean value of the
function ~ = ~(t), and B(s, t), s, t E T, is its correlation
function:*
a(t) = M~(t),
B(s, t) = M[~(s) - a(s)][~(t) - a(t)], s, t E T. (2.10)
Therefore, the Gaussian measure P on a a-algebra ~~ can be defined
uniquely by means of its mean value a(t), t E T, and its
correlation function B(s, t), s, t E T.
The mean value a(t), t E T, can be arbitrary, and the correlation
function B(s, t), s, t E T, need only satisfy the positive
definiteness condition
n
for any tb ... , tn E T and real Cb ... , Cn'
(2.11)
For any function a(t), t E T, and a positive definite correlation
function B(s, t), s, t E T, there exists a Gaussian random function
with the mean a(t), t E T, and a correlation function B(s, t), s, t
E T. Actually, Gaussian distri butions Pt " ... , tn with the mean
[a(tl), ... , a(tn)] and correlation matrices {B(tb t j)} are
compatible distributions, and define a Gaussian measure P in the
space X = \RT of all real functions x = x(t) of t E T, on the
a-algebra m = ~~, which can be generated by the given values ~(t) =
~(x, t) on X of the form (2.8) (parameter t runs through the set
T).
I.3 Lemmas on the Convergence of Gaussian Variables
Let ~n = ~n(w), n = 1,2, ; .. , be a sequence of nindom variables
on a prob ability space (Q, ~,P). The sequence ~n' n = 1, 2, ... ,
is said to be conver gent in probability on a set A E ~ to some
variable ~ = ~(w) if for any e > 0
lim p({I~n - ~I > e} n A) = O. (3.1) n-+ DO
Let us recall that a sequence ~n' n = 1,2, ... , converges in
probability if and only if this sequence is Cauchy, t i.e., on the
same set A the sequence .1 nm = ~n - ~m' n, m = 1, 2, ... ,
converges to zero in probability.
Lemma 1. If a sequence of Gaussian variables ~"' n = 1, 2, ... ,
converges in probability on a set A E ~ of positive measure (P(A)
> 0), it is convergent in the mean:
lim M[~n - ~y = O. (3.2) n-+ DO
* M~ denotes the expectation of a random variable ~ = ~(Q)) on a
probability space (Q, W, P):
M~ = fn ~(w)P(dQ)). t See, for example, [10], p. 90.
5
I Preliminaries
Proof. We shall consider Gaussian variables Ll nm = ~n - ~m' For
any 8 > 0
P{ILlnml > 8} = 2 fXJ;' exp { 2n(Jnm
where anm = MLl nm, (J;m = P(Llnm - anm)2. Suppose that the
sequence ~m n = 1, 2, ... , is not convergent in the mean; this is
equivalent to
lim (a;m + (J;m) > O. n, m- 00
It can be easily seen that under this condition for a positive 8 we
have
lim P{ ILlnml > 8} ~ 1 - p/2, n, m-+ 00
where p = P(A) > O. But then
lim P( {ILlnml > 8} n A) ~ p/2, n, m- 00
which fact contradicts (3.1). Hence
lim MLl;m = lim (a;m + (J;m) = 0, n, m-+ 00 n, m-+ oc
i.e., the sequence ~n' n == 1, 2, ... , is Cauchy (a fundamental)
in the mean and, therefore, is convergent in the mean.
In particular, if a sequence of Gaussian variables ~n' n = 1, 2,
... , is convergent with positive probability (i.e., convergent for
all w from a set A E 'll of positive measure), it is convergent in
the mean. D
Let us consider a sequence of independent Gaussian variables ~n' n
= 1,2, ....
Lemma 2. The series I:= 1 ~; is convergent with positive
probability if and only if the series I:=1 M~; is convergent.
Proof. Obviously, 00 00
and hence the convergence of the series I:= 1 M~; implies that the
variable ~2 = I:= 1 ~;(w) is finite for almost all WE Q, i.e., the
series I.% 1 ~; is con vergent with probability one. Let the
series I:= 1 ~; be convergent with positive probability (by the
well-known zero-one law* this series 'is con vergent with
probability one as well). Then the sequence ~n' n = 1, 2, ... ,
converges to 0 in the mean: M~; ~ 0 for n ~ CX) (see Lemma 1). Let
an =
* See, for example, [10], p. 157.
6
1.4 Gaussian Variables in a Hilbert Space
M~n' a; = M(~n - anf. Then
a2 + a 2 = M()")2 + i x 2 _1_ exp{ n n 'on Jlxl> 1 J2iian
where the random variables ~~ = ~~(w) are defined as
~~(w) = {o~n(W) for I~nl !( 1 for I~nl > 1.
For a; + a; -40 we have
so that M(~~)2 ~ a; + a;.
By the well-known three-series theorem* a necessary condition for
the series 2:;;,,= 1 ~; of independent variables ~;, = 1,2, ... ,
to be convergent is that 2:;;,,= 1 M(~~)2 < 00. But M(~~)2 ~ a;
+ a;, and, consequently, it follows from the convergence of the
series 2:;;,,= 1 ~; that 2:;;,,= 1 (a; + a;) < 00. 0
1.4 Gaussian Variables in a Hilbert Space
A random variable ~ in a Euclidean n-dimensional space IRn is said
to be Gaussian if its probability distribution is Gaussian.
The random variable ~ E [Rn is Gaussian if and only if the real
variable ~(u) = (u, ~) (equal to the scalar product ofthe elements
u, ~ E [Rn) is Gaussian for each u E [Rn.
In fact, the value at a point u E [Rn of the characteristic
function cp(u) of the random variable ~ E [Rn coincides with the
value of the characteristic function of the real random variable
~(u) = (u, ~) at the point 1 and has the form
cp(u) = Mei(u,~) = eXP{i(U, a) - ~ (Bu, U)}. U E IRn
(see (Ll), where (u, a) is the mean and (Bu, u) is the variance of
the Gaussian variable ~(u) = (u, ~)).
It is clear that the random variable ~ E [Rn is Gaussian if and
only if the random function of the form ~(u) = (u, ~) of u E [Rn is
Gaussian.
Let U be a complete separable Hilbert space and let ~ = ~(w) be a
func tion on a probability space (Q, m:, P) with the values in U.
The random element ~ of a Hilbert space U is said to be a random
variable in U if the scalar product (u, ~) for each u E U is a real
random variable, i.e., it is a measurable function on the
probability space (Q, m:, P).
* See, for example, [10J, p. 166.
7
I Preliminaries
The random variable ~ in a Hilbert space U is said to be Gaussian
if the real random variable ~(u) = (u, ~) is Gaussian for each u E
U. This fact is equivalent to the fact that the random function
~(u) = (u, ~) of u E U is Gaussian since values ~(u) = (u, ~) as
well as any vector values [~(Ul)' ... , ~(un)] are Gaussian.
In fact, for any vector A = [Al' ... , An] in [Rn the scalar
product Lk= 1 Ak~(ud is equal to
Jl Ak~(Uk) = Ctl AkUb ~) = (u, ~), where u = Lk= 1 AkUk E U; by
hypothesis, the variable ~(u) = (u, ~) is Gaussian.
Obviously, the mean
a(u) = M(u, ~), UE U,
of the random function ~(u) = (u, ~), U E U, is a linear
functional, and the correlation function
B(u, v) = M[(u, ~) - a(u)][(v, ~) - a(v)], u, V E U,
is a bilinear positive functional on the Hilbert space U. In this
case, since the scalar product (u, ~) is a continuous function of u
E U for each fixed WE Q, a Gaussian function ~(u) = (u, ~) of u E U
must be continuous in the mean (see Lemma 1):
lim M[(u, ~) - (v, ~)]2 = 0 (4.1) Ilu - vll-+ 0
(Ilull denotes the norm of the element u E U). But
M[(u, ~) - (v, m2 = a(u - V)2 + B(u - v, u - v)
and (4.1) implies that the functionals a(u) and B(u, v) are
continuous. Being a linear continuous functional, the mean a(u) can
be expressed as
a(u) = (u, a), UE U, (4.2)
for some element a in U. Any element a E U having the property
that
(u, a) = In (u, ~(w))P(dw) (4.3)
for all u E U is said to be the mean* of a random variable ~ E U.
Being a continuous positive bilinear functional, the correlation
function B(u, v) can be expressed
B(u, v) = (Bu, v), U, VE U, (4.4)
where B is a linear positive (i.e., nonnegative self-adjoint)
operator in a Hilbert space U called a correlation operator.
Let us show that the correlation operator B is completely
continuous.
* For the integrability of functions with values in a Hilbert
space, see, for example, [12], p. 59.
8
1.4 Gaussian Variables in a Hilbert Space
In fact, any orthonormalized sequence Vb V2, ... goes to zero
weakly, so that the Gaussian variables ~n = (vn' ~), n = 1,2, ... ,
where ~ = ~(w) E U, goes to zero as n -4 00 for all WE Q. Therefore
(see Lemma 1), they are convergent in the mean, i.e.,
M~; = (Bvn' Vn) -4 0
(here and further on we assume for simplicity of notation that the
mean a E U is 0). If the operator B was not assumed completely
continuous, outside some e-neighborhood of zero there would be an
infinite number of spectral points (taking into account the
multiplicity), and, therefore, an infinite number of invariant
orthogonal subspaces for each element of which
(Bu, u) = r U(Elu, u);:' ellul12, Jlll>e
where B = SA dEl is the spectral representation of the continuous
self adjoint operator B.
Further, we shall choose a complete orthonormalized basis of eigen
elements Vb V2, ... of this completely continuous symmetric
positive opera tor B corresponding to eigenvalues ai, a~, ... ,
The corresponding variables ~k = (Vb ~), k = 1,2, ... , are
uncorrelated:
In this case 00 00
for j = k,
for j # k.
L ~r(w) = L (Vb ~(W))2 = 11~«(())112. 1 1
As is well known, uncorrelated Gaussian variables are independent
and the convergence of the series Lr' ~r(w) (for all w) implies the
convergence of the series Lr' Ma (see Lemma 2). Consequently,
00 00 00
L (BVb Vk) = L M~r = L ar < 00, 1 1 1
i.e., the correlation operator B is a nuclear operator:* for any
orthonormal system Ub U2, .•. , E U,
(4.5)
Therefore, if we have a Gaussian random variable ~ E U, the random
function ~(u) = (u, ~) of parameter u E U has a mean of the form
(4.2) and a correlation function of the form (4.4) where the
correlation operator B is a nuclear operator on the Hilbert space
U.
Next, let ~(u), u E U, be an arbitrary Gaussian random function
with a mean of the form (4.2) and a correlation function of the
form (4.4), where B
* See, for example, [7J, p. 55.
9
I Preliminaries
is a nuclear operator on the Hilbert space U. Then there exists an
equivalent random function ~(u), u E U, and a Gaussian random
variable ~ = ~(w) in U such that
~(u) = (u, ~), UE U. (4.6)
The variable ~ E U indicated can be defined for almost all
elementary outcomes w by the formula
00
~(w) = L ~(VdVb (4.7) k=l
where Vb V Z, ... is the complete orthonormal system of
eigenelements of the nuclear operator B, and, by virtue of the
relation
00 00
M L ~(Vk)Z = L B(Vb Vk) < 00 k=l k=l
for independent Gaussian variables ~(vd, ~(vz), ... , the series
Lk'= 1 ~(Vk)2 is convergent with probability one. In fact, ~(u), u
E U, is a random linear functional in the sense that with
probability one
~(A1Ul + A2U2) = Al~(Ul) + )~2~(U2)
for any real Ab A2 and any elements Ub U2 E U since, as we can
easily verify,
M[~(A1Ul + A2U2) - Al~(Ul) - A2~(U2)J2 = O.
Furthermore, the random functional ~(u) is continuous in the mean
(see (4.1) and below), and, since
n
we have
~(U) = !~~ ~ Ct (u, V)Vk) = !~~ ktl (u, Vk)~(Vk) (in the sense of
convergence in the mean); at the same time with probability
one
(u,~) = !~~ Jl (U, Jl ~(Vk)Vk) = !~~ ktl (u, Vk)~(Vk)' so that we
have the equality (4.6) with probability one for each value ~(u) of
the primary random function ~(u), U E U.
Thus, we have arrived at the following result.*
Theorem 1. The Gaussian functional ~(u), U E U, on a Hilbert space
U can be represented by (4.6) if and only if the mean a(u), U E U,
is a continuous
* A survey of results related to distributions in linear spaces can
be found, for example, in Yu. V. Prokhorov, 'The method of
characteristic functionals," Proceedings of the 4th Berkeley
Symposium, Vol. 2,1961, pp. 403-419.
10
1.4 Gaussian Variables in a Hilbert Space
linear functional, and the correlation function B(u, v), u, V E U,
is a con tinuous bilinear functional, the corresponding operator B
being a nuclear operator in the representation (4.4) of the
bilinear functional B(u, v).
As is well known, a Hilbert space U can be identified with the
adjoint space X of all linear continuous functionals on U. In fact,
each functional x E X can be defined uniquely by the formula x =
(u, x), U E U, where x E U is a fixed element of the space U.
Following the exposition in Section 2, any Gaussian random function
~(u), U E U, of the form (4.6) corresponds to a probability
distribution p~ in a Hilbert space X = U (sample functionals ~(u) =
(u, ~(w)), U E U, belong to X). The Gaussian measure p~ is defined
on the rr-algebra 5B generated by cylinder sets ofthe space X = U
of the form
[(Ul' x), ... , (un, x)] E r, (4.8)
where U b ... , Un E U and r are Borel sets in an n-dimensional
Euclidean space; the rr-algebra 5B is generated, obviously, by
variables of the form
~(x, u) = (u, x), XEX, (4.9)
where the parameter u runs through the space U = X. For any element
a E U and a positive nuclear operator B there exists a
Gaussian random function ~(u), u E U, of the form (4.6) with a mean
a E U and a correlation operator B. There also exists a Gaussian
function of this type with values defined by (4.9) on the
probability space (X, 5B, P~).
Example. Gaussian Variables in the Function Space ;t?2(T). Let ~ =
~(t) be a Gaussian random process on an interval T of the real line
with mean a(t), t E T, and let correlation function B(s, t), s, t E
T, satisfy the following con dition: for all s, t E T,
lim [a(s) - a(t)] = 0, s-->t
(4.10) lim [B(s, s) - 2B(s, t) + B(t, t)] = 0. s-->t
This condition implies that the random process ~ = ~(t) is
continuous in the mean:
lim M[~(s) - ~(t)]2 = 0. s-->t
As is well known, * in this case there exists an equivalent
measurable process (with values ~(t) = ~(w, t)) such that the
function ~ = ~(w, t) with respect to a pair of variables (w, t) on
the product of spaces Q x T is measurable. We shall consider that
the initial Gaussian process ~ = ~(t) itself is Gaussian. Assume
that the condition
ST B(t, t) dt < 00, (4.11)
* See, for example, [10J, p. 209.
11
By Fubini's theorem on iterated integration
IT M~2(t)dt = In IT e(w, t)dtP(dw) < 00
and almost all sample functions ~(w, .) = ~(w, t) of t E T belong
to a Hilbert space 22(T) of real square-integrable functions u =
u(t) of t E T with the scalar product
(u, v) = IT u(t)v(t)dt.
We redefine the values ~(w, t) for those w E Q for which the sample
func tions ~(w, .) = ~(w, t), t E T, do not appear in 22 (the set
of such WE Q has measure 0) and go over to a measurable Gaussian
process ~ = ~(t) whose sample functions belong to the Hilbert space
22. This random function ~ = ~(w, .) can be treated as a random
element in the Hilbert space 22.
Consider the scalar products
(u, ~) = IT u(t)~(w, t)dt, UE u.
Since ~ = ~(w, t) is a measurable function of the variables (w, t),
for each fixed u E U the real function (u, ~) of WE Q is also
measurable, i.e., is a random variable. Therefore, ~ = ~(w, .) is a
random variable in a Hilbert space 2 2(T). By Fubini's theorem the
random function ~(u) = (u, ~) of u E U has the mean
a(u) = IT u(t)a(t)dt = (u, a),
and the correlation function
UE U,
B(u, v) = IT IT u(s)v(t)B(s, t) ds dt = (Bu, v), u, VE U,
where the correlation operator B is given by a kernel B(s,
t):
Bu(t) = IT B(s, t)u(s) ds.
(4.12)
(4.13)
The random variable ~ E 22 is Gaussian. In fact, as can be easily
seen, for a continuous function u = u(t) of t E T the random
variable (u, ~) = S T u(t)~(t) dt is the limit in the mean of
Gaussian variables of the form
n
L U(tk)~(tk)(tk - tk - 1), k=l
where to ~ tl ~ ... ~ tn are subdivision points of an interval T
and for an arbitrary function u E 22 the variable (u, ~) is the
limit in the mean of Gaussian variables (un' ~) (here Un = un(t), n
= 1, 2, ... , is a sequence of
12
1.5 Conditional Probability Distributions and Conditional
Expectations
continuous functions convergent in the mean to the function u =
u(t)). The limiting value for a sequence of Gaussian variables is,
as known, also Gaussian.*
I.5 Conditional Probability Distributions and Conditional
Expectations
Let ~(u) = ~(w, u), u E U, be a family of Gaussian random variables
on a probability space Q and let m-(U) be the rr-algebra of sets in
Q generated by all variables ~(u) = ~(w, u) on Q (parameter u runs
through U). We shall assume for simplicity that M~(u) = 0, U E U.
Denote by H(U) the Hilbert space of random variables '1 (measurable
with respect to the rr-algebra m-( U)) with the scalar
product
(5.1)
Denote by H(U) the closed linear hull of all variables ~(u), u E U.
Let S and T be some subsets in U.
We shall consider an arbitrary variable '1 E H(S) and the
projection fi of it onto a subspace H(T). Since H(U) is the
aggregate of Gaussian variables, the difference ,1 = '1 - fi as a
Gaussian variable orthogonal to all variables ~(t), t E T, is
independent of all these variables. Therefore,
'1=fj+,1, (5.2)
where fj is the variable described above, measurable with respect
to the rr-algebra m-(T), and ,1 is a Gaussian variable with zero
mean and variance rr2 = M('1 - fi)2; this Gaussian variable is
independent of ~(t), t E T. It can be easily seen t that the
conditional distribution ofthe variable '1 with respect to the
rr-algebra m-(T) always exists and that for almost all w it is a
Gaussian distribution with the mean
M{'1Im-(T)} = fi(w) (5.3) and the constant variance
(5.4)
Let us recall that the conditional expectation M{'1Im-(T)} is
geometrically the projection of the variable '1 E H(U) onto the
subspace H(T), and in our case this is the variable fj, i.e., the
projection onto the subspace H(T).
It is well known that Gaussian variables have finite moments of all
orders. Denote by Hn( U) the closure of the subspace of all
variables of the form
(5.5)
where CP(Xb ... ,xd is a polynomial of degree not higher than n in
the arbitrary number of variables Xb' .. , Xb and U b ... , Uk E
U.
* See, for example, [10]. p. 33.
t See, for example, [6], p. 75.
13
I Preliminaries
Theorem 2. For any variable I] E Hn(s) its conditional expectation
q =
M{I]I~l(T)}belongs to the subspace Hn(T) (with the same index
n):
M{I]I2l(T)} E Hn(T). (5.6)
Proof. We can consider without loss of generality that the set S
and the set T are finite (say, S = {st. ... , Sk} and T = {tt. ...
, td). In fact, we can go over to the general case by passage to
the limit, * since
I] = lim I]m' M{I]I2l(T)} = lim M{lIml2l(T)}, m->oo
m->oo
where the convergence is in the mean, and
and also
m->oo
Let I] = q{r;(Sl), ... , ((Sk)], where <p(xt. ... , x k) is a
polynomial of degree not higher than n. As we have seen, the
theorem is certainly true for n = 1. Assume that it is true for all
indices not exceeding n - 1. Denote by ~(s) the projections of
variables ((s), j = 1, ... , k, onto the subspace H(T). The
differences ((Sj) - ~(Sj),j = 1, ... , k, are independent of ((t),
t E T. Set
(= <p[r;(Sl) - ~(Sl)' ... , ((Sk) - ~(sdJ.
The variable ( is independent of ((t), t E T, and in the
expansion
k 8 ~ I] - (= j~l 8xj <P[((Sl), ... , ((Sk)]((S) + ...
the right-hand side is a linear combination of expressions of the
form
<Pm[((sd, ... , ((Sk)] 'l/In-m[~(Sl)' ... , ~(sd],
where <Pm(xt. .. . , Xk) and l/In-m(x 1 , ••• ,xd are
polynomials of degree not higher than m and n - m, respectively,
with m ~ n - 1. By hypothesis the conditional expectation fim = M
[l]ml2l( T)] of the variable 11m = <Pm[ ((s d, ... , ((sd]
belongs to the subspace Hm(T), m ~ n - 1. Obviously, the product
fim . l/In-m[~(sd, ... , ~(Sk)] belongs to Hn(T) and hence also the
linear com bination of these products, the conditional expectation
of the difference I] - ( equal to fi - M( being such a linear
combination. Therefore, fim =
M[I]I2l(T)] E W(T), as was to be proved. D
We shall define next Hermite polynomials in several variables. Let
P(dx) be Gaussian measure in a k-dimensional space IRk of
vectors
x = [Xl' ... , Xk] and let H be a Hilbert space of all real square
integrable
* See. for example [6], pp. 29, 287.
14
1.5 Conditional Probability Distributions and Conditional
Expectations
functions cP = cp(x) of x E IRk with the scalar product
< cp, 1jJ> = f~k cp(x)ljJ(x)P(dx).
As is well known* a Gaussian distribution has finite moments of all
orders, and the aggregate of all polynomials q> = CP(Xb ... ,
Xk) of the vari ables Xb' .. ,Xk is a set everywhere dense in H.
Any polynomial cp = CP(Xb ... , xk) of degree p orthogonal to all
polynomials of degree lower than p will be said to be a Hermite
polynomial.
Denote by H p the aggregate of all Hermite polynomials of the same
degree p. Obviously, H p is a finite-dimensional subspace, and the
Hilbert space H is the sum of orthogonal subspaces H P' P = 0, 1,
... :
Let us consider a Gaussian vector variable [~(ud, ... ,~(Uk)].
Denote by H p(Ub ... , Uk) the aggregate of all variables of the
form
11 = cp[ ~(Ul)' ... , ~(Uk)]'
where q> = CP(Xb ... , Xk) is the Hermite polynomial of degree p
with respect to the distribution P of the vector variable [~(Ul)'
... ,~(Uk)]. Obviously
00
(5.7)
Lemma 3. For any Sb ... , Sk and tb' .. , tz, andfor any p and q
(with p =1= q),
the subspaces Hp(Sb"" Sk) and Hq(t 1, ••• , t l ) are
orthogonal:
Hp(Sb' .. , sd ~ Hq(t b ... , t l ). (5.8)
Proof. Let p < q for definiteness. We shall consider an
arbitrary variable 11 E Hp(Sb .... , Sk) and the conditional
expectation q = M{I1IU(t1, ... , t l )} of this variable. By
Theorem 2 the variable q belongs to the subspace HP(t b ... , t l )
= I~=o EB Hr(tb ... ,tl ). But q is the projection of the variable
11 onto the whole space H(tb ... , t l ) so that the difference 11
- ij is orthogonal to H(tb ... , t l ) and, in particular, 11 - ij
-.l Hq(t b ... ,tl ). At the same time ij ~ Hq(t b .. . , t l ),
since ij belongs to HP(t b ... , t l ), orthogonal to Hq(t b ... ,
tl )
for p < q. Consequently,
11 = [(11 - ij) + ij] ~ Hitb ... , t l ),
which was to be proved. o * This fact can be explained as follows.
Obviously, the system of functions of the form eil!, x>, (t, x)
= 2)kXx, is complete in a complex space H, and
I N [i(t x)]"112 0- 21"+ 1) eil!, x) - I --' ,- ,;; c -( 1)' -+ 0,
where 0- 2 = f (t, X)2 P(dx),
"~[ n, n + ,
I Preliminaries
Furthermore, we shall define the subspace H p( U) as the closed
linear hull of all subspaces Hp(u b ... , ud where Ul , ... , Uk E
U (index p is the same for all Ub ... ,ud. Obviously, by virtue of
(5.7) and (5.8) we have
n 00
Hn(u) = l: EB Hp(U), (5.9) p=o p=o
where Ho(U) contains only constant variables; Hl(U) = H(U) is the
closed linear hull of variables ~(u) E U; H z(U) is the closed
linear hull of variables [~(Ul)~(UZ) - B(ub uz)], where Ub Uz E U;
H3(U) is the closed linear hull of variables [~(ud~(UZ)~(U3) -
~(udB(uz, U3) - ~(UZ)B(Ul' U3) - ~(u3)B(Ub uz)] where Ub Uz, U3 E
U; etc.
As seen from (5.9), the conditional expectation M [l1l~(T)] of any
variable 11 E H p(S) with respect to the a-algebra ~(T) belongs to
the subspace H p( T) (with the same index p).
We recall the general formula for products of Gaussians:
M~(Ul) ... ~(un) = l:TI B(Ub Uj), (5.10)
where the sum is to be taken over all subdivisions of the set (u b
... , un) into pairs (Uk, Uj ), and the product is to be taken over
all pairs (Ub Uj) of the corresponding subdivision.
We obtain this formula from the relation
on M~(ud ... ~(Un) = OAl ... OAn cp(O),
where cp(A) = M exp{i(A, ~)}, the characteristic function of the
Gaussian vector ~ = [~(ud, ... , ~(un)]' is of the form (see
(1.1))
I.6 Gaussian Stationary Processes and the Spectral
Representation
A Gaussian random process ~(t) = ~(w, t) with values in a
probability space Q, where the parameter t takes integer (discrete)
or real values ( - 00 < t < 00), is said to be stationary if
its mean is constant
a(t) = M~(t) == a
and the correlation function B(s, t) depends on the difference (s -
t) only:
B(s, t) = M[~(s) - a][~(t) - a] = B(s - t) (6.1)
(in what follows we shall take a = 0). The function B(t) in (6.1)
is said to be a correlationfunction ofthe stationary
process ~(t); it can be expressed as
B(t) = f eiAtF(dA), (6.2)
1.6 Gaussian Stationary Processes and the Spectral
Representation
where F(dJ.) is called the spectral measure of the stationary
process ~(t)
(positive bounded measure). In (6.2) the integration is over - n
,;:; J. ,;:; n in the case of discrete "time" t and over - 00 <
). < 00 in the case of continuous time t.
The stationary process ~(t) permits a spectral representation of
the form
~(t) = f eiAtcI>(dJ.), (6.3)
where cI>(d),) is called the stochastic spectral measure such
that
McI>(,11)cI>(,12) = F(Ll 1 n Ll 2)'
Each variable '1 from the closed linear hull H(T) of the values
~(t), t E T, permits a spectral representation of the form
1] = f q>(J.)cI>(d)'), (6.4)
where q>(J.) is the function from the space LT(F), the real
linear hull of the functions eiAt of )., t E T, closed with respect
to the scalar product
(6.5)
The stochastic integral given by (6.4) is defined for any function
q> E LT(F) and yields '1 E H(T). The correspondence
1]+--+q>().) is a unitary isomorphism* of the Hilbert spaces
H(T) and LT(F):
(1]1> '12) = (q>b q>2)F' (6.6)
In the case where the parameter t is continuous and the set T is a
finite interval, we can define the space LT(F) as the closure of
the subspace L 0
(by the scalar product given by (6.5)) of all functions of the
form
q>().) = fTeiAtu(t)dt, (6.7)
where the u = u(t) are infinit-ely differentiable functions
vanishing outside of the interval T. Since the functions q>(J.)
decrease faster than IJ.I-n as J. -+ 00,
the scalar product (6.5) can be defined on the subspace L 0 with
the help of a finite spectral measure as well as any (J-finite
measure G(dJ.) satisfying the condition
for some integer n. Let us set
(6.8)
and define the complete Hilbert space LT(G) to be the closure of
all functions of the form (6.7) by the scalar product given by
(6.8).
Let LT(G) be a Hilbert space of the type indicated. (6.4)
prescribes the random functional '1 = '1(q» defined on the
everywhere dense subspace of
* For this see, for example, [24].
17
I Preliminaries
functions LT(G) n LT(F). We want to know under what conditions ry =
ry(cp) is (to within an equivalence) a random element from the
conjugate space of LT(G), i.e., a Gaussian linear continuous
functional on the Hilbert space LT(G).
Suppose ry = ry(cp) is a random element from the conjugate space of
LT(G), I.e.,
(6.9)
where ry = ry(A) is a Gaussian function with trajectories in the
Hilbert space LT(G). The correlation operator B can then be found
from the relations
<BCPb C(2)G = Mry(cpdry(CP2) = <CPb C(2)F = <Acpb Ac(2)F =
(A*Acpb C(2)G,
where A is the operator on the Hilbert space LT(G) into the Hilbert
space LT(F) determined by the equality
Acp(A) = cp(A), (6.10)
and where A* is its adjoint on LT(F) into LT(G). B will be a
nuclear operator like any correlation operator.
On the other hand, if the operator B = A * A is nuclear by Theorem
1 the Gaussian linear functionalry = ry(cp) with correlation
functional <Bcpb C(2)G is equivalent to a Gaussian element in
the space adjoint to LT(G) determined by (6.9).
Let us note that we have the inclusion
LT(G) s:: LT(F)
not only for a nuclear operator but also for any bounded operator B
= A * A, since
Ilcplli = <Acp, ACP)F = <Bcp, cp)G:::; IIBII·llcpll~· We note
that for the finite measure G(dA), (6.9) is equivalent to a
spectral
representation of the initial stationary process ~(t), t E T:
t E T. (6.11)
In fact, the functions cp(A) = eiAt are complete in the Hilbert
space LT(G) and, from (6.11), ry(eiAt) = <e iAt , ry)G, t E T,
extends to the whole space LT(G), the closed linear hull of
functions of the type cp(A) = eiAt.
If we want to deal with the initial random process ~(t), t E T,
rather than its functionalry(cp), cP E LT(G), we need to introduce
the space X of all real functions x = x(t), t E T, permitting a
spectral representation as
x(t) = f e-iAtljJ(A)G(dA), t E T, (6.12)
where IjJ(A) E LT(G) and the values of x(t) coincide with values of
the linear continuous functional < cP, IjJ)G with cp(A) = eiAt•
It is seen that (6.12) provides the one-to-one correspondence
between x E X and IjJ E LT(G).
If we introduce a scalar product so that
<Xb X2) = <1jJ1> 1jJ2)G, (6.13)
18
1.6 Gaussian Stationary Processes and the Spectral
Representation
where t/!l and t/!2 correspond to Xl and X2, respectively, X will
become a complete Hilbert space.
If we consider the special case where G(dA) = (1/2n) dA and the
Hilbert space LT(G) consists of functions of the form
(6.14)
where X = x(t) belongs to the usual L 2(T) space of real
square-integrable functions with the scalar product
<Xl, x 2) = fT Xl(t)X2(t)dt,
(6.14) yields the usual Fourier transform. By the well-known
Plancherel formula,
which implies that if(6.11) and (6.12) are understood in the sense
ofa Fourier transform, the Hilbert space X of all square-integrable
functions X = x(t), t E T, is seen formally as a special case of
the general scheme of construction of a Hilbert space with scalar
product given by (6.13).
We have actually proved the following:
Theorem 3. A random process ~(t), t E T, is (to within an
equivalence) a random element of a Hilbert space X if and only if
the product B = A * A is a nuclear operator on a Hilbert space
LT(G) where the operator A is defined by (6.10).
We shall prove below that for a rather wide class of absolutely
continuous measures F(dA) and G(dA) with densities f(A) = F(dA)/dA
and g(A) = G(dA)/dA, the operator A * A is nuclear if
f f(A) g(A) dA < 00 (6.15)
(there exist many cases where the operator A * A is not nuclear if
the condition given by (6.15) is not satisfied).
We shall examine in Chapter III an operator of the form L1 = A! A 1
- E where A! is an operator of the same type as A but which maps
LT(G) onto a space LT(Gl ) constructed with respect to the measure
Gl(dA) having density gl(A) = g(A) + f(A).
Since f(A) = gl(A) - g(A), we have
<L1cp, t/!)G = <A!AlCP, t/!) - <cp, t/!)G
= <cp, t/!)G, - <cp, t/!)G
19
I Preliminaries
for any 4>, tf; E L T ( G); therefore, the operator A * A
coincides with LI. We prove (see Chapter III, Theorem 17) that the
condition
f [I(A)]2 g(A) dA < ro
is sufficient for the operator LI = A * A to be a Hilbert-Schmidt
operator. Hence, if we introduce the space LT(F 1) with 11 (A) =
-/I(A)g(A), the operators of the type indicated,
B, B2 LT(G) ~ LT(F 1) and LT(F d ~ LT(F),
will be such that B!Bl and B!B2 are Hilbert-Schmidt operators under
the condition given by (6.15). It can be easily seen that A*A =
B!B!B2Bl is nuclear (see, for example, [7J, p. 57).
I.7 Properties of the Sample Functions*
1.7.1 Differentiability in the Mean: Some Asymptotic
Relationships
Let ~(t), - Cf) < t < ro, be a Gaussian stationary process
with continuous time t.
A process ~(t) is said to be differentiable (in the mean) if there
exists a limit (in the mean)
This limit exists if and only if in the Hilbert space LT(F), where
T =
( - ro, ro), the limit eiAh - 1 . .
lim e'At = iAe'At h~O h
exists; obviously, this is equivalent to the condition
f.':'oo A 2 F(dA) < ro (7.1)
(here F(dA) is the spectral measure of the stationary process ~(t».
If
~(t) = f_oooo eiAt<P(dA)
is the spectral representation of the Gaussian stationary process
~(t), its derivative ~'(t), - ro < t < ro, being also a
Gaussian stationary process will be
~'(t) = f.':'oo eiAt(iA)<P(dA).
It can easily be seen that the condition given by (7.1) is
equivalent to
(7.2)
20
I. 7 Properties of the Sample Functions
as h ~ 0, where B(t) is the correlation function and .d n denotes
the operator of taking the difference (.dhB(t) = B(t + h) - B(t)).
In fact, by virtue of (7.2) we have for any A that
fA ),2 F(dA) ~ C fA 1 - cos Ah F(dA) ~ C .d - h.d hB(O) - A -- - A
h2 "" h2
for sufficiently small h (no matter how large the prescribed A is)
where C is a constant; hence (7.1) follows from (7.2).
Let us consider a nondifferentiable stationary process ~(t) and
examine the conditions on the spectral measure F(dA) for which we
have the relation
(7.3) where 0 < rJ. < 1.
We have
.d - h:z:B(O) = f~oo C -~~dh J (1 - codh)l -a F(dA)
~ C f~oo A2aF(dA);
f~oo A2a F(dA) < 00. (7.4)
We discuss in more detail the case where there exists a spectral
density f(A) = F(dA)jdA. For sufficiently large A, IAI ?: A,
let
f(A) = IAI-P
(where f3 > 1 since the spectral density f(A) must be an
integrable function). For a non differentiable stationary process
we must have f3 ~ 3. Let f3 < 3. Then
Substituting the variable Ah = f.1 we have*
f oo 1 - cosAh dA = hP-2a-1 foo 1 - cosu du A APh2a Ah uP
~ hP-2a-1 foo 1 - cosh du Jo uP
and, therefore,
where
* IX - f3 for the variables IX and f3 implies that lim IX/ f3 =
1.
21
I Preliminaries
It can easily be deduced from the relations thus obtained that
if
f(}.) = O{I}.I- P},
(7.3) will be satisfied for 20( = f3 - 1; if
lim f(}.)I}.I P = 00, .l.~eo
we shall have for 20( = f3 - 1 that
-I' - Ll_hLlhB(O) _ 1m hZ'" - 00. h~O
(7.5)
(7.6)
Similarly, for f3 = 3 we obtain for a spectral density f(}.) of the
type given by (7.5)
L1_h~~B(O) ~ c reo 1 - ~osu du + 0(1) = O{llnlhll}, JAh u
and, therefore, (7.7)
1.7.2 Continuity
Let us consider the nondifferentiable Gaussian stationary process
~(t), - 00 < t < 00, with correlation function B(t)
satisfying (7.3).
Theorem 4. Under the condition given by (7.3) there exists an
equivalent Gaussian process ~(t) for each trajectory of which, for
sufficiently small h,
(7.8)
uniformly with respect to t in each finite interval, where C is a
constant.
Proof. For sufficiently small h we have
P{ILlh~(t)1 > c'lhl"'llnllhII1/2}
= _2_ r e- x2/z dx ~ J(c'/c"Jllnlhll ' /2
~ _2_ r xe-x2/2 dx "" J2ir Jrc'lc"Jl1nlhll ' /2
= _2_ e1/2(c'lc"J21nlhl = -2-lh I P
J2ir J2n ' 1 (C')2
f3 = 2. c" '
where c" is the constant in the relation Ll-hL1hB(O) ~ c"lhl z",
and c' is chosen so that f3 > 1.
22
I.7 Properties of the Sample Functions
Consider the initial process ~(t) at binary-rational points of the
form t = kl2n, assuming for simplicity that 0 ,,:; t ,,:; 1. For h
= 2 -n we have
Since f3 > 1 and the series I:= 1 2 -(p-l)n is convergent, it
follows from the Borel~Cantelli lemma that for sufficiently small
h
uniformly for k = 0, ... , 2n - 1. It can easily be seen that any
interval [kI2n, kd2"'] can be taken as the
sum of intervals [rI2m, (r + 1)j2m] where rand m are integers, but
not more than two intervals of the type indicated for any m.
Therefore, for any h (with appropriate n within r" ,,:; h ,,:; 2
-n+ 1)
where I* denotes the summation over corresponding m. Taking this
fact into account, we get for sufficiently small h
,,:; I* c'r"mlln 2m11/2 m
,,:; Qhl"llnlhlll/2.
Therefore, the trajectories of the process ~(t) considered satisfy
with prob ability 1 the condition given by (7.8) on the set of all
binary-rational points; in particular, for almost all WE Q the
trajectories ~(w, .) = ~(w, t) are uni formly continuous functions
on the set dense everywhere of binary-rational points tkn .
Obviously, for an arbitrary point t the limit limtkn _ t ~(w, t kn
)
exists and coincides with the initial value of ~(t) = ~(w, t) for
almost all W E Q. It is seen that the trajectories of the
equivalent process with values defined as
~(W, t) = lim ~(w, tkn)
23
I Preliminaries
satisfy the condition given by (7.8) for almost all w. Defining the
variables ~(w, t) for the remaining w (assuming, for example, ~(w,
t) == 0), we obtain an equivalent process ~(t) whose trajectories
satisfy the condition given by (7.8). The theorem is proved.
0
1.7.3 Limit Theorems
Let us consider a nondifferentiable Gaussian stationary process
~(t) with correlation function B(t). Assume that on an interval (0,
T) the second deriv ative B"(t) exists (with the exception ofa
finite number of points) with discon tinuities of the first kind
only, i.e., the finite limits B"(t - 0) = limh-+o B(t - h) and B"(t
+ 0) = limh-+ o B(t + h) exist within the interval (0, T) for any
point t. We recall (see Section 1.1) that for the nondifferentiable
process ~(t)
1. LI_ hLi hB(O) 1m h2 = 00;
h-+O (7.9)
therefore, the derivative B"(t) has discontinuities of the second
kind at the point t = O. More precisely,
lim B"(h) = - 00. h-+O
Theorem 5. We have the following limiting relation:
. 1 N-l [Llh~(kh)J2 hm- I = 1, h-+O N k=O LI-hLlhB(O)
(7.10)
where h = TIN and the limit is understood in the sense of the mean
convergence.
Proof. Let n be a number of points of discontinuity of B"(t). Let
us take an arbitrarily small 8 > O.
Each point of discontinuity t, 0 ~ t ~ T, can be enclosed in some
interval so that the total length of these points does not exceed
8. Let us denote by IE the complement of the union of these
intervals (IE is the union of a finite number of intervals). It is
clear that the function B"(t) is uniformly con tinuous on the set
IE and
LI_hLlhB(t) = O{ B"(t)h2}
uniformly with respect to t E Ie. Taking into account (7.7) we
have
LI_hLlhB(t) = o{ LI_hLlhB(O)}
(7.11)
(7.12)
uniformly with respect to t E Ie. Furthermore, since the function
LI_hLlhB(t) as well as the correlation function B(t) are positive
definite, it follows that for all t
(7.13)
24
We have
and, by (5.10),
M[Llh~(S)' Llh~(t)· Llh~(U)' Llh~(V)] = Ll-hLlhB(s - t)· Ll-hLlhB(u
- v) + Ll-hLlhB(s - u) . Ll_hLlhB(t - v)
+ Ll-hLlhB(s - v) . Ll_hLlhB(t - u).
It is easy to calculate that
(J (h) = M - I-I Z [1 N-l Llh~(kh)Z JZ N k~O Ll-hLlhB(O)
=~ NIl [Ll_hLlhB((k-j)h)JZ. N k,j~O Ll-hLlhB(O)
(7.14)
For fixed j no more than 1 + [bh- l ] points of the form (k - j)h
can belong to each interval of the length b; the total number of
such points does not exceed N(l + [bh- l ]). Therefore, the number
of points of the form (k - j)h in the complement of the set I, does
not exceed the number (Nn + NZc/r:). It follows from (7.11)-(7.14)
that for any c > 0, for sufficiently small h
where C is a constant. The theorem is proved. D
Let us note that there exists a subsequence hi> h2' ... , for
which the limiting relation given by (7.10) is satisfied with
probability 1. Furthermore, we can take as the subsequence hI, hz,
... any sequence for which
(7.15)
since under the condition given by (7.15) there exists a sequence
Cn ~ 0 such that
{ 1 1N-I[Llh~(khW I } I P - I-I ~ Cn < 00, h~hn N k~O
Ll_hLlhB(O)
and it follows from this by the Borel-Cantelli lemma that we have
conver gence with probability 1 in (7.10) for h = hn • It is
interesting to examine the question of the rate (defined in (7.14))
at which the function (J2(h) de creases as h ~ 0:
25
I Preliminaries
Lemma 4. Under the assumptions made before on the function B"(t) we
have the following estimate:
N-l
(7,16)
Proof. It is seen that
d2(h) = 0 {h- 1[Ll_ hLl hB(0)]2 + h2 II [B"(s - t)]2dsdt},
Is-tl>2h
Further, for any fixed b > ° II [B"(s - t)]2 ds dt = 0 {I:h
B"(t)2 dt},
Is-tl>2h and if the function B"(t) is monotone in a certain
neighborhood (0, b) then
I:h B"(t? dt = B"(2h + 8)[B'(b) - B'(2h)],
where 2h ~ 8 ~ band
B"(2h + 8) = O{h2Ll_ hLl hB(0)},
If the function B"(t) is monotone, it retains the sign in some
neighborhood (0, b); therefore, B'(t) is also monotone. Obviously,
the function LlhB(t) =
S~+h B'(s) ds as well will be monotone. Hence, when iB'(2h)i--+ 00
as h --+ 0, then
iB'(2h)i = O{h-1[LlhB(h)]},
Therefore,
II [B"(s - t)]2dsdt = 0 {h- 2i Ll_ hLl hB(0)i, if B'(2h) is bounded
Is-tl>2h h- 3 iLl -hLl hB(OW, if B'(2h) is unbounded.
As a result we have
d2(h) = O{max(ihi-liLl_hLlhB(OW, iLl-hLlhB(O)i)}·
The relation given by (7.16) yields the following estimate for the
function (J2(h):
(7.17)
In particular, under the condition given by (7.3) the function
(J2(h) de creases as h --+ ° at least as ihip:
f3 = max{l, 2(1 - ct}}.
26
1.7 Properties of the Sample Functions
Therefore the condition given by (7.15) will be satisfied, for
example, for any sequence of the form hn = 2 -n, n = 1, 2,
....
For the type of stationary Gaussian processes considered, we have
finally the following result, completing Theorem 5.
Theorem 6. Under the condition that
L1-hL1hB = 0{lhI 1/ 2 }
we have the limiting relation
(7.18)
1 N-1
lim h -1 - I L1h~(kh)L1h~(t + kh) = B'(t - 0) - B'(t + 0), (7.19)
h~O N k=O
where t is any fixed point of the interval (0, T), N = [h -1( T -
t)] - 1, and convergence is in the mean.
Proof. Elementary calculations similar to those carried out before
yield that the variable
has the mean
(h) = L1hL1-hB(t) = B(t - h) - B(t) _ B(t + h) - B(t) MI1 h h
h'
its variance being such that
DI1(h) ~ Cd2(h).
It is seen from the estimate given by (7.16) for the variable d2(h)
that we have (7.19) under the condition given by (7.18).
Of course, the relation in (7.19) holds with probability 1 for a
sequence h = hn' n = 1, 2, ... , decreasing sufficiently rapidly.
D
27
11.1 Preliminaries
Il.l.l Introduction
We have already seen (in Section I.6) that the Hilbert space of
random variables H(T) generated by a stationary process ~(t), t E T
(with spectral measure F(d)')), is isometric to a space of
functions LT(F), which is the closed linear hull of the functions
eiM for). E [ - n, n] in the case of discrete time t
and for)' E [ - 00, 00] in the case of continuous time t. This fact
enables us to investigate stationary processes using analytic
tools. To do this, it is useful first to study in detail the
analytic structure of spaces LT(F), as we shall do in this chapter,
restricting ourselves to the case where T is a finite interval or a
half-line.
It is clear that we may limit ourselves to the interval T = [ - T,
T] or T = [0, T] and half-lines T = (- 00,0] or T = [0, (0), since
an arbitrary interval or the half-line Tl can be obtained by a
"shift" of T by some real t, and the corresponding space LdF) is
obtained by multiplying LT(F) by eiAt •
To make it easier for the reader to grasp the results to be
obtained in this case, we shall assume that ~(t) is a stationary
process with discrete time with spectral density f().) = 1. It is
seen that here LT(F) consists of trigo nometric polynomials P(e
iA) = LIE T c(t)e iM if T is a finite interval, and in fact
coincides with the Hardy space £,2 within a circle (or outside of a
circle) if T is a half-line; more precisely, LT(F) consists of
square-integrable func tions q>().) that can be expanded in a
single Fourier series:
o q>().) = L c(t)eiAI for T = ( - 00, 0]
-00
28
cp(A) = L c(t)eiAt for T = [0, 00)).
° What is the space LT(F) if f(A) i= 1 in the case of continuous
time
processes? It is useful to note initially that if the spectral
measures F(dA) and G(dA)
are related by the inequality
F(dA) ;:::: G(dA)
(F(dA) majorizes G(dA)), the corresponding Hilbert spaces LT(F) and
L T( G) satisfy the inclusion
This is immediate from the fact that any fundamental sequence of
func tions of the form CPn(A) = Lk CkneiAtkn, n = 1, 2, ... , in
the space LT(F) con vergent to a function cp(A) E LT(F), is, at
the same time, fundamental in the space L T ( G):
for m, n ~ 00. In this case the limiting function I/J(A) E LT(G)
coincides al most everywhere relative to G(dA) with the limiting
function cp(A) E LT(F) indicated above. Therefore, the functions
cp(A) and I/J(A) coincide, being the elements of the Hilbert space
L T( G), i.e., cp(A) E L T( G).
This fact immediately implies that if we have a spectral density
f(A) = F(dA)/dA of the type*
f(A) >< 1 (1.1)
(in the case of discrete time t), as well as a spectral density
f(A) = 1, the space L[o, t](F) consists of all the
polynomials
t
t=O
(with real coefficients c(t), ° ~ t ~ r). Then the spaces L(-oo,
o](F) (respec tively, L[o, oo)(F)) consist of square-integrable
functions that can be expanded as a Fourier series of the
form
° cp(eiA) = L c(t)eiAt
-00
that coincide with boundary values (for r ~ 1) of analytic
functions cp(z) in the circle Izl < 1 (outside of a circle) of
the Hardy spaces £2 mentioned above:
cp(eiA) = lim cp(z), r .... 1
* We recall that for the variables a and P the relation a x P
implies that 0,;;; C 1 ,;;; alP,;;; C2 < 00.
29
(1.2)
where n is an integer, is an analog of the condition given by
(1.1). For n = 0 the space L[D. tiF) defined in Section 1.6
consists obviously of
square-integrable functions <p(Je) that can be expressed as
Fourier integrals of the form <p(Je) = So eiAtc(t) dt.
Similarly, the spaces L( - 00, D](F) (respectively, L[D, oo)(F))
consist of functions of the form <p(Je) = S~ eWc(t) dt
(respectively, <p(Je) = So eWc(t)d(t)).
Making use of elementary knowledge only, we can show that under the
condition given by (1.2) the space L[o, t](F) coincidls with the
class of functions that can be expressed as
<p(Je) = P(iJe) + (1 + iJet Sot eiAtc(t)dt, (1.3)
where P(iJe) is a polynomial of degree less than n - 1 (with real
coefficients) and c(t) is a square-integrable (real)
function.
In fact, all the functions <p(Je) = (iJe)ke iAS, k = 1, ... , n
- 1, being limits of the form
eiA(s + h) _ eiAS <p(Je) = lim (iJe)k-1 h '
h~D
belong to the space L[D, tiF), so that each polynomial P(iJe) =
L(j-l ck(iJe)k is contained in L[D, t](F). Furthermore, the
functions <p(Je) = (1 + iJet- 1
(e(l + iA)S - 1),0 :::; s :::; r, contained in L[o, t](F) can be
expressed as
where
<p(Je) = (1 + iJet f~ eWcs(t) dt,
{ et for 0 :::; t :::; s, cs(t) = o for s :::; t :::; r.
It can be readily seen that the closed linear hull of functions
<p(Je) = (1 + iJe)"-I(e(1 +iA)S - 1), 0:::; s :::; r, and
<p(Je) = (iJe)\ 0:::; k :::; n - 1, forms the whole space L[D,
t](F) (being, by definition, the closed linear hull of functions
eiAS, 0:::; s :::; r) since, starting from the functions <p(Je),
we can arrive at the functions cp(Je) = eiAS by means of iterated
integration:
f~ (1 + iJer-Ie(l+iA)sds = (1 + iJe)n-2(e(1+iA)t - 1);
etc. It can also be seen that the linear hull of "step" functions
cs(t) of the type indicated, in which the parameter s runs through
the entire interval [0, r], is everywhere dense in the Hilbert
space :;e2[0, r] of square-integrable functions c(t), 0 :::; t :::;
r. Moreover, for functions <p'(Je) and <p"(Je) of the
form
<p(Je) = (1 + iJe)n f~ eiAtc(t) dt,
30
ILl Preliminaries
where c(t) is a linear combination of all step functions cs(t), we
have by the well-known Parseval equality
Ilcp'(Je) - cp"(Je)lli = f~CXJ Icp'(Je) - cp"(JeWf(Je) dJe
x f~CXJ Icp'(Je) - cp"(JeWI1 + iJel 2n dJe
= 2rc f; IC'(t) - c"(tW dt.
It is clear that the closed linear hull of all functions cp(Je) of
the form indicated coincides with the class of functions cp(Je)
that can be expressed as
cp(Je) = (1 + iJe)n f; eiAtc(t) dt,
where c(t) E 2'2[0, ,]. Adding to this all the functions <p(Je)
= (iJe)\ ° ~ k ~ n - 1, we obtain, obviously, the space L[o.
T](F).
(1.3) enables us to describe adequately the variables in the space
H(T) for T = [0, ,], the closed linear hull of variables ~(t), ° ~
t ~ ,. In fact, if cJ>(dJe) is the stochastic spectral measure
of a stationary process ~(t), every variable 1] E H(T) can be
represented as an integral 1] = S cp(Je)cJ>(dJe) (see Section
1.6) in which cp(Je) E LT(F). It is easily seen that
1] = :t: [ak~(k)(O) + bk~(k)(,) + f; ~(k)(t)Ck(t) dt}
where ak and bk are real coefficients, ck(t) are square-integrable
functions, and ~(k)(t) are derivatives in the process, k = 0, ... ,
n - 1.
Let us note that if the spectral density f(Je) satisfies the
condition
f(Je) ~ c(1 + Je2)-n
only, the corresponding space L[o. TiF) is contained in the space
LT(G) associated with the spectral density g(Je) = c(1 + Je2)-n.
Hence each func tion cp(Je) E L[o. T](F) can be represented by
(1.3). It is also crucial to note that (1.3) determines an entire
analytic function for all complex Je. Later on (see Section IlI.4)
we shall show that the space L[o. TiF) can be identified with the
class of functions of the form (1.3) under the condition given by
(1.2), as well as under the weaker condition
(i.e., f(Je) x (1 + Je 2)-n for sufficiently large Je only). Under
the condition given by (1.2) we can easily deduce from the
representation (1.3) the general formula for functions <p(Je) in
the spaces L(_ CXJ, o](F) and L[o. CXJ)(F). Every function cp(Je) E
L[o, CXJ)(F) is, in fact, the limit of a sequence of functions
CPk(Je) E L[o, Tk](F), 'k ~ 00, that can be expressed as
k = 1,2, ... ,
II The Structures of the Spaces H(T) and LT(F)
where the sequence Ck(t), k = 1, 2, ... , is fundamental in a
Hilbert space £,2(0, 00) of square - integrable functions and
convergent to a function c(t), ° ~ t ~ 00, in this space. It is
seen that the limiting function <p(Je) =
limk~w <Pk(J,) can be expressed as
<peA) = P(iJe) + (1 + iJet foOO eiltc(t) d(t),
where P(iJe) = lim Pk(iJe) is a polynomial of degree less than n -
1. It is also seen that this function (where c(t) E £,2(0, 00))
belongs to the space L[o, oo)(F).
Similarly, the space L( _ 00, o](F) coincides with the class of all
functions that can be described by the formula
<p(Je) = P(iJe) + (1 + iJet f~oo eWc(t)d(t).
We shall investigate further the structure of the corresponding
spaces LT(F) when the spectral density f(Je) does not necessarily
satisfy the condition given by (1.2) but still vanishes at infinity
although "not too rapidly," i.e.,
f" lnf(A) dA > - 00 for discrete time t and
f oo lnf(Je) dJe > - 00 -00 1 +Je 2
for continuous time t. In this case, retaining the earlier
notation, we con sider complex spaces H(T) and LT(F).
II.l.2 Functions Analytic in a Circle
Let us denote by YfP, 1 ~ p ~ 00, the class of analytic functions
<p(z) in a unit circle Izl < 1 for which
If <P E YfP, then for almost all Je E [ - n, nJ the boundary
values of <p(eil) = limr~ 1 <p(reil) exist and
The space YfP is a Banach space with a norm 11<pII(P) = (S~"
1<p(eil)iP dJe)l/P. We can identify YfP with the closed subspace
(in the known space* £,P( - n, n)) of all functions <p(eil ) E
£,P( - n, n) for which
f:" <p(eil)einl dJe = 0, n = 1,2, . .. .
* The space £,P(a, b) consists of functions <p(A) on the
interval a ~ A ~ b for which 11<pII(P) =
m I <p(AJ!P dA)'/p < 00.
32
ILl Preliminaries
We shall denote by yep the given subspace consisting of boundary
values of the functions described analytic in a circle.
A function cp(z) analytic inside the circle Izl < 1 is said to
be an outer function if we can represent it as
{ 1 f'" eiA + z } cp(z) = aexp -2 -.-, -lnp(A)dA , n -'" e'A -
z
lal = 1,
where the real function p(A) is nonnegative and In p E 2l( - n, n).
A function cp(z) analytic inside a circle is said to be an inner
function
if Icp(z)1 ::;; 1 and Icp(eiA)1 = 1 for almost all A E [ - n, n J.
By the Blaschke product we shall mean an analytic function B(z) of
the
form
[ ~ IX - ZJPn B(z) = azP Il 11X:11 ~ ~nZ ' lal = 1,
where p, Pi> P2, ... are nonnegative integers, 0 ::;; IlXnl <
1, and the product IlllXnlpn is convergent.
Theorem ([15], pp. 98-99). An inner function cp(z) is uniquely
representable as the product
cp(z) = B(z) exp - '" -.-, - Jl(dA) , { f eiA + z } -'" e'A -
z
where B(z) is a Blaschke function and Jl(dA) is a singular
measure.
It follows readily from this result ([15], p. 123) that any
nonempty family of inner functions has a greatest common (inner)
divisor.
We shall denote by D the class of functions cp(z) (introduced by V.
I. Smirnov-see [23]) analytic in a circle Izl < 1 that permit
the representation
{ f '" eiA + z } { 1 f'" eiA + z } cp(z) = B(z)exp - -.-, -Jl(dA) x
exp -2 -.,-lnp(A)dA, (1.4) -'" e'A - z n -'" e'A - z
where B(z) is the Blaschke product, Jl(dA) is a singular measure,
and p(A) ~ 0, lnp E 2l( -n, n). Therefore, class D consists of
functions cp(z) that can be expressed as the product of some inner
function (the inner part of cp) and some exterior function (the
outer part of cp).
For each function cp(z) of class D there exist, for almost all A,
boundary values cp(eiA) = limr~ 1 cp(reiA) satisfying the condition
Icp(eiA) I = p(A), where p(A) is the function from the
representation given in (1.4) for cp(z).
Theorem ([15], p. 80). Each function cp E yel is the product of two
functions from ye2.
Theorem ([15], p. 81). For f(A) > 0 and fE gl we have f = Icpl2
where cp E ye2 if and only if lnfE 21.
33
II The Structures of the Spaces H(T) and LT(F)
With respect to this theorem, we note that one can take
cp(z) = exp - -.-, - lnf().) d)' . { 1 f" eiA + z } 4n -" e'" -
z
Beurling's Theorem ([15], p. 145). For cp E Ye 2 the functions
{zncp}, n = 0, 1, ... , generate the total class Ye 2 if and only
if cp is an outer function.
Let cp(z) be a function analytic in a circle. In this case the
function cp(l/z) is analytic outside of a circle. Associating in
particular each function analytic in a circle with a function
analytic outside a circle, we have classes D and YeP of functions
analytic outside of a circle. To distinguish between these two
classes, we denote, if necessary, by D+ and Yep+ the classes inside
a circle and by D- and YeP- those outside a circle.
II.].3 Functions Analytic in a Half-Plane
Let us denote by yfP and D the classes of functions analytic in the
upper half-plane that are images of classes YeP and D in a circle
under the conformal mapping of the circle onto the upper
half-plane.
We shall denote by YeP the class of functions cp(z) analytic in the
upper half-plane for which
f~oo Icp(x + iy)lp dx :;::;: M < 00, y ~ 0,
where the constant M is independent of y ([15], [17]). A function
cp E D is said to be outer if it can be expressed as
cp z =exp - ~-~-). ( ) { I foo 1 + )'z In p().) d } ni - 00 ). - z
1 + ).2 '
where p().) is a real function, p().) ~ 0, and Inp().)/(1 + ).2) E
21( - 00,00).
We have equality for outer functions in the inequality for
Poisson's integral:
Inlcp(z)1 :;::;: l' foo lnlcp().)I d)', n - 00 (). _ X)2 + y2 z = x
+ iy, y > 0.
A function cp E D is said to be inner if Icp(z)1 :;::;: 1 and
Icp().) I = 1 (z = ). + if.1, f.1 ~ 0).
We have for functions of class D representations of the type
(1.4):
. { foo 1 + )'z } {1 foo 1 - )'z In p().) } cp(z) = e"XZB(z)exp -
-occ ). _ z f.1(d)') x exp ni -(0). _ z 1 + ).2 d)' ,
where ()( is a real number, B(z) is a Blaschke function, f.1(d)')
is a singular finite measure, and p().) ~ 0, Inp()')/(1 + ).2) E
21( - 00,00).
34
IL2 The Spaces L + (F) and L - (F)
The other assertions can be generalized to classes D and yep in the
upper half-plane in a similar way. In particular, we have the
following:
Theorem of Lax ([15J). For cp E ye2 the functions {eiAtcp(A), t:;?;
O} generate all ye2 if and only if cp is an outer function.
From now on the following characteristic feature of the spaces ye2
will be frequently used.
Paley-Wiener Theorem ([15J, p. 187). Thefunction cP E ye2 (in the
upper half plane) if and only if
cp(z) = fooo eiztc(t)dt, Imz:;?; 0,
where c(t) E 'p2(0, (0).
If we consider classes D and yep in the upper half-plane and
classes D and yep in the lower half-plane at the same time, we
shall write D + and yep + in the case of the upper half-plane and D
- and yeP- in the case of the lower half-plane. Note that
'p2( - 00, (0) = ye2 + EB ye2 - .
II.2 The Spaces L +(F) and L -(F)
Let ~(t) be a stationary wide-sense process with a spectral measure
F(dA). Let F = Fa + F., where Fa is the absolutely continuous
component and Fs is the singular component of the measure F. Set
f(A} = dFa/dk In this case we shall call f(A) a spectral density
even if F #- Fa. For the sake of brevity, let us introduce some new
notation:
L = L(F) = L(-oo, oolF), L - = L -(F) = L(_ 00, o](F),
L + = L +(F) = L[o, oo)(F).
Theorem 1. If ~(t) is a stationary random process with discrete
time t = 0, ± 1, ... , then:
1. L + (Fs) = L -(Fs) = L(Fs); 2. L + (Fa) = L -(Fa) = L(Fa) if and
only if
f" lin f(A) I dA = 00;
3. if f~" lin f(A) I dA < 00, (2.1)
35
II The Structures of the Spaces H(T) and LT(F)
we can write f(A) as f(A) = Ig(eiAW where 9 is an outer function of
class ye2 in the circle Izl < 1.
Further,
Theorem 2. If ~(t) is a stationary random process with continuous
time t, - 00 < t < (f), then:
1. L +(Fs) = L -(Fs) = L(Fs); 2. L + (Fa) = L -(Fa) = L(Fa) if and
only if
f oo Ilnf(A)1 dA = (f)'
-00 1+A2 ' 3. if (2.2)
f oo Ilnf(A)1 dA -001+A2 <00,
we can write f(A) as f(A) = Ig(),W, where 9 is now an outer
function of class ye2 in the upper half-plane 1m z > 0, z = A +
ill.
Further,
Assertions (1)-(3) of the two theorems are, in essence, equivalent
to those due to A. N. Kolmogorov and M. G. Krein in the theory of
prediction of stationary random processes. * Since these assertions
play a fundamental role from now on we shall sketch the proofs
briefly.
Proof. Let us first prove (1)-(3) of Theorem 1. Suppose that F is a
singular measure on an interval [ - n, n J, and that L - #- L. Then
necessarily eiA 1= L - . Denote. by cp().) the projection of the
element eiA on the subspace L -. Then eiA - cp(A) =1= 0 and eiA -
cp(A) 1. L -, and
f" einA(eiA - cP)F(dA) = 0, n = 0, 1, . .. .
Since the generalized measure Fl(dA) = (eiA - cp)F(dA) is analytic,
that is,
n = 0,1, ... ,
by the theorem of F. and M. Riesz ([15J, p. 73) it must be
absolutely con tinuous with respect to a measure F(dA). The
contradiction obtained proves (1).
As to (2), suppose J~" lIn f(A) I dA = (f). Assume contrariwise
that L - #- L. Then we have again that eiA 1= L -, and if cp(A) is
the projection of eiA onto
* See. for example, [24].
f:" ein2t/1(l)f(A)dA = 0, n = 0,1, .... (2.3)
Let us denote by 11<pII(p) the norm in the space 'pP( -n, n). We
note that the function t/lf E 'p1( -n, n). In fact, by the Schwarz
inequality,
I I t/lf I 1(1) ~ 11t/lIIFa . (1Ifll(1))1/2 < 00,
from which it follows by (2.3) that t/lf E yt'1. The logarithm of
any function from yt'1 is summable; therefore,
J:" lnlt/l(A)f(A) I dA > - 00.
Further, from the elementary inequality In x < x we have
f:" lnlt/l 2(A)f(A) I dA ~ Iit/lilia < 00,
f:" In f(l)dA ~ J:" f(A)dA < 00.
Together with the ptevious inequality the above two inequalities
imply lnf E .p1, contrary to the hypothesis.
The contradiction obtained proves the first part of (2); the second
part of (2) will follow from (3), which we Shall prove next. By
virtue of (2.1) we can write f(A) as f(l) = Ig(ei2W, where
{ 1 f ei6 + Z } g(z) = exp 4n :" ei8 _ z lnlf(8)1 d8 ,
is the exterior function from yt'2. Let <p(ei2) = <p E L
+(F). This fact implies that there exists a sequence of polynomials
Pn(z) such that 11<p - PnllF ---+ 0. But in this case it follows
also that II<pg - P ngll(2) = 114> - P nilF ---+ ° for n ---+
00.
Obviously, Png E yt'2+. Hence the limiting function t/I = <pg E
yt'2+, that is, <p = t/J/g where t/I, 9 E H2 +. Taking the
canonical representation given in (1.4) of the functions from yt'2+
and D+, we can see that <p E D+.
Conversely, let <p E D+ n L(F). Then t/I = <pg E yt'2+. The
function 9 is exterior, and by Beuding's theorem (see Section 11.1)
the aggregation of functions {gP} in which P runs through all the
polynomials is dense in yt'2 + .
This fact implies, in particular, that we can find a sequence of
polynomials Pn for which, as n ---+ 00, we have
that is, that <p E L +(F). The case L -(F) should be considered
in a similar way. The proofs of (1)-(3) of Theorem 2 coincide
almost completely with the
proofs of (1)-(3) of Theorem 1. In fact, we can prove with the aid
of the
37
II The Structures of the Spaces H(T) and LT(F)
conformal mapping of a circle onto a half-plane that the
generalized theorem of F. and M. Riesz is true in this case as well
(see [23J, p. 209). Further, if cP E £"2 in the upper half-plane,
then, necessarily, we have
Finally, in proving (3) we should refer to the Lax theorem instead
of the Beurling theorem. 0
It is useful to note that, in proving (3) in Theorem 1 and Theorem
2, we obtain, in essence, the following: if the conditions given by
(2.1) or (2.2) are satisfied, then
II.3 The Construction of Spaces LT(F) When TIs a Finite
Interval
(2.4)
We investigated in Section 11.2 the spaces LT(F) (and therefore,
the spaces H(T) generated by values of the corresponding stationary
process ~(t),
t E T, with the spectral measure F(dA)), when T is an infinite
interval. We shall investigate in this section the case where T =
[a, b J is a finite interval.
Since the case of discrete time is trivial-LT(F) consists of trigon
ometrica I polynomials of the form Ia~t~b ateiAt-we shall deal only
with processes with continuous time. Furthermore, we restrict
ourselves to the investigation of a process ~(t) with an absolutely
continuous spectral measure F(dA) and a spectral density f(A)
satisfying the condition given by (2.2). As noted above, it
suffices to consider intervals of the form T = [ - a, a]. Let
L"(F) = n LT(F) T=[-a,a],a>"
and
The space L O(F) is defined by the behavior of the process ~(t) in
an infinitely small neighborhood of zero; its isometric space HO =
nTH T, in particular, contains all existing derivatives
~(k)(O).
Let us agree to denote by D" the aggregate of entire analytic
functions cp(z), z = A + ill, of finite degree ~ (J, i.e., entire
functions such that
lim R -1 max Inlcp(ReW)1 :( (J
R---+oo (J
(in particular, Do denotes the aggregate of entire functions of
zero degree).
38
11.3 The Construction of Spaces LT(F) When T Is a Finite
Interval
Theorem 3. If the spectral density f(A) of the stationary process
~(t) satisfies the condition given by (2.2), then* it follows
that
L"(F) = D" n L(F), L O(F) = Do n L(F). (3.1)
Proof. We need to prove two inclusions:
L"(F) c D" n L(F) and L"(F) ~ Du n L(F).
We shall prove the first inclusion for all (J ~ 0, and we shall
prove the second inclusion for (J = ° only. t
1. Proof of the inclusion L"(F) c D" n L(F). Given any function (fJ
E L"(F), there exist functions
(fJn(A) = I ajnexp{itjnA}, j
such that 11(fJ - (fJnlIF < lin, n = 1,2, .... It is seen that
all (fJn E Du+ lin' Next let us prove that at each point of the
complex half-plane we have
(3.2)
for any Ii > 0, the constants C, being dependent of Ii only (and
not of n). The uniformly bounded family of analytic functions (fJn
is compact.
Furthermore, II(fJn - (fJIIF ~ 0, and hence (fJiz) is convergent to
an entire function (fJ(z) which, under the condition given by
(3.2), is a function of finite degree less than (J. It can be seen
that the restriction of (fJ(z) on 1m z = ° coincides with
(fJ(A).
Therefore, we need only to prove the inequality in (3.2). To this
end let us estimate I (fJn(z) I along 45 degree rays through the
origin, and make use of the Phragmen -Lindelof principle. t
We shall estimate first l(fJn(Z) I for Ifll ~ 1. We note that, by
virtue of (2.2), f(A) = Ig(AW, where g(z) is an exterior
function of class ye2. Let us introduce the functions t/ln(z) =
(fJng exp{ iZ«(J + 6)},
* We do not distinguish between the functions from D. and the
restriction of them on the line !l = O. t The proof of the general
case (0" > 0) is similar but rather cumbersome-the reader can
find this proof in N. Levinson and H. McKean, "Weighted
trigonometrical approximation on [Rl with Application to the germ
field of a stationary Gaussian noise, Acta Math. 112, Nos. 1-2
(1964), 99-143. A stronger result was obtained earlier by M. G.
Krein who gave the integral representation for entire functions
from U(F)-see "On the basic approximation problem of the theory of
extrapolation of stationary random processes," DAN SSSR 94 (1954),
13-16.
The further results of this chapter are borrowed from the paper by
Levinson and McKean cited above; we have altered their proofs
slightly.
I See, for example, [20].
39
II The Structures of the Spaces H(T) and LT(F)
0< (5 ~ lin. It is clear that l/Jn E ye2, and therefore for all
J1 > 0 we have
f-"'oo Il/Jn()' + iJ1W dJ1 ~ f-"'oo ll/Jn(JeW dJe
= I/CPnl/~ ~ (I/CP/IF + ~y ~ (I/CP/IF + 1)2 = C1 . (3.3)
Further, since the function 9 E ye2 it can be represented by the
Paley Wiener theorem as
z = A + iJ1, J1 > 0,
where g(u) is the Fourier transform of the function g(A). Hence for
all J1 > 0
(3.4)
Denote by r R the contour consisting of a segment of a line IAI ~
R, J1 = t, and an arc resting on this segment of a circle with
radius R centered at the point z = i/2, Re z ;,: t. By the Cauchy
formula, if Re Zo ;,: t, and the radius R is sufficiently large, we
have
Relying on the relation given in (3.4) we can easily show that the
integral over the semicircle r R vanishes starting from l/J(z)!(z -
zo) as R ~ 00. Con sequently, by virtue of the inequalities given
in (3.3) for all J1 ;,: 1 we obtain
. 1 00 Il/Jn (u + ~)I ll/Jn(Je + IJ1)1 ~ 2n f-oo 1 . (1 )1 du
(u - A) + I - - J1 2
1 (foo 1 ( i)12 foo du )1/2 ~ 2n - 00 l/J n u + "2 du - 00 u2 + ~ =
C 3 · (3.5)
Next, writing Inlg(z)1 as the Poisson integral (g is an exterior
function, see Section II.l)
Inlg(z)1 = ~ foo Inlg(u)1 du, n - 00 (u - A)2 + J12
z = A + iJ1,
we find that if z = Rei6, () = n14, 3n14, then, as R ~ 00,
40
Inlg(z)1 ~Izl-~O. (3.6)
11.3 The Construction of Spaces LT(F) When T Is a Finite
Interval
In fact, for a fixed T > 0 and z = Rei6 we have
~ fT Inlg(u)1 du = 0 (~) = 0 (~) --+ O. n -T(u-;,l+/12 /1 R
At the same time we have
/11 r Inlg(u)1 d I /1 2 + ;,.2 + 1 r Ilnj(u) I d
;Jlul>T(U-)"f+/12 u~ 2n/1 Jl ul>T1+u2 U
= ~ R2 + 1 r Ilnj(u)1 du n.}2 R2 Jlul>T 1 + u2
= R . 0(1), T --+ 00,
which proves (3.6). It follows from (3.6) that for any c > 0 and
z = Rei8 (0 = n/4, 3n/4; R ;:: 1)
the inequality Ig(z)l;:: C4e-elzl is satisfied, where the constant
C4 might depend on c. Noting (3.5) we can obtain the estimate
(3.7)
on the rays z = Rei9 (0 = n/4, 3n/4; R ;:: 1). Similarly, if we
introduce the function t/ln -(z) = cpngexp{ -iz(a + o)},
where g(z) = g("2) E yt"2 - for z from the lower half-plane, we
shall obtain the estimate
(3.8)
on the rays z = Rei8 (0 = 5n/4, 7n/4; R ;:: 1). We shall take next
the estimate of ICPn(z)1 on segments of the rays lying
within a circle Izl ~ 1. We assume, as usual, that In + a = In a if
a > 1, and In + a = 0 if a ~ 1. Estimating the subharmonic
function Inlcpn(z)1 with the aid of the Poisson integral, we have
on the lines 11m zl = 1 that
lnlcpn(z)I ~ ~ foo Inlcpn(u)1 du n - 00 (u - ),,)2 + 1
1 u2 + 1 foo In + ICPn(u)1 d ~ ~ sup u
n u (u-),,)2+1 -00 u2 +1
~ ~ ),,2 + 2 foo In+lcpn(u)1 du n 2 - 00 u2 + 1
In + ICPn(u)llg(u)1
= ~ ),,2 + 2 foo Ig(u)1 du n 2 - 00 1 + u2
41
II The Structures of the Spaces H(T) and LT(F)
Therefore, the function e- C7z2 CPn(z), analytic in a strip 11m
zl:( 1, is bounded in this strip and satisfies on its boundaries
the inequality le- C7Z2 cpn(z)1 :( Cs, where Cs is independent of
n. According to the Phragmen-Lindelof prin ciple the last
inequality holds true at each point of the strip indicated. In
particular, all the functions CPn(z) are uniformly bounded in the
circle Izl :( 1 From this, and from (3.7) and (3.8), it follows
that the inequality given in (3.2) holds true, which proves, as
mentioned above, the first part of the theorem.
2. Proof of the inclusion L o(F) => Do n L(F). Let cp(),) E Do n
L(F). By the familiar Hadamard theorem* the function cp(A), the
entire function of zero degree, can be written as the product
(3.9)
where the Zn =I- 0 are zeros of cp(z). The function cp(z) can be
rewritten as the sum CPl (z) + IPz(z), where cP l(Z) = i(cp(z) +
cp( - z)) and cpz(z) = i(cp(z) - cp( - z)) are even and odd,
respectively. In this case, we have as before that CPt. cpz E Do n
L(F). Therefore, we need only to prove the theorem for even
functions and odd functions. In both cases the proofs are
identical, and for the sake of definiteness we shall consider the
even functions cp(A).
We need to prove that for any 8 > 0 there is a function CPo E
U(F) for which Ilcp - lPellF :( 8.
Let us note first that each square-summable entire function cp(A)
of finite degree :( 8 belongs to U(F). In fact, each function of
this kind belongs to L(F). By the Paley-Wiener theorem t the
function cP has, relative to entire functions from !£'Z, the
Fourier transform rp equal to zero outside of [ - 8, 8 J. We infer
that
and is finite, IP E U(F). Therefore, it suffices to construct a
square-summable integral function of
degree :( 8 adequately approximating cpo We note that the Hadamard
fac torization given by (3.9) for the even function cP will be of
the form
m~O.
We shall define the function lPo(A) by the equality
CPo(A) = AZm n (1 -A:) I1 (1 _ AZ~Z), IZnl<d Zn n>db n
where () = lOin; the number d = d(8) will be defined later
on.
* See, for example, [20J, p. 525.
t See, for example, [15J, p. 82.
42
11.3 The Construction of Spaces LT(F) When T Is a Finite
Interval
We shall show next that CfJe is a square-summable integral function
of degree::::; G (and, therefore, CfJe E L 2(F)). The Euler
formula
00 ( 22) sin n2 = n2 If 1 - n2
enables us to write CfJe as
(3.10)
Therefore CfJe(2) is an entire function of degree nb = G. We
estimate next the ratio of polynomials in the right-hand side of
(3.10) for large 2. Let us introduce the monotone nondecreasing
function N 'P(R) equal to the number of roots of the function
CfJ(z) in a circle Izl ::::; R. The function N 'P(R) is closely
related to the growth order of the function CfJ(z). In particular,
N CfJ(R) = oCR), R ~ 00,* for function
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