International Conference Probability and Statistics with ... · Probability and Statistics with Applications ... computer science and informatics. ... International Conference Probability

University of Debrecen

Faculty of Informatics

International Conference

Probability and Statistics with

Applications

Abstracts

Dedicated to the 100th

anniversary of the birthday of

BÉLA GYIRES

June 8-12, 2009

Debrecen, Hungary

Dedicated to the 100th anniversary of the birthday of Bela Gyires

Bela Gyires1909–2001

Emeritus professor of Debrecen University and doyen of the Hungarianmathematicians Bela Gyires passed away in Budapest on August 26, 2001, atthe age of 92. His research activity embraced matrix theory, linear algebra,probability theory and mathematical statistics. He belonged to those few out-standing mathematicians who deliberately raised no strict boundary betweenpure and applied mathematics. He left us about hundred papers. His nicest,most important and best known results were in linear statistics. Many of hisresults are now cited in monographs and textbooks.

Bela Gyires was born on March 29, 1909 in Zagrab (Zagreb). His fatherwas a clerk of the Hungarian State Railways. After the first World War hisfamily moved to Debrecen, where he finished secondary school. He gradu-ated at the Peter Pazmany University at Budapest and got his Ph.D. degreefrom the Jozsef Nador Technical University, Budapest. In his reminiscenceshe considers as his master Charles Jordan (Jordan Karoly). After the sec-ond World War in 1945 he got post at the side of Prof. O. Varga in theMathematical Seminar (Chair) of Debrecen University. He was promoted toa private-docent in 1946, and from that time for 55 years he served our Uni-versity and became a renowed professor of it. He not only witnessed thedevelopment, which led from a two-person-staff Seminar to a big Institute ofMathematics and Informatics with seventy highly qualified mathematicians,but he was the most active creator, and for a long period the director ofthis institution. He was the founder and for thirty years the head of the De-partment of Probability and Applied Mathematics. He had a decisive role inincluding in the university-curriculum such highly important modern fields asprobability theory, mathematical statistics, computer science and informatics.He was a highly esteemed and much liked teacher of generations of mathe-matics students. His fate granted him not only a long, but also an activelife lasting until the last hours. His book: Linear approximations in convexmetric spaces and the application in the mixture theory of probability theory,written upon request, appeared when he was 84. At the time of his death hehad still accepted but not yet appeared papers, and who knows how manyplans in his mind.

The Hungarian Academy of Sciences elected him in 1987 as a correspond-ing, and in 1990 as an ordinary member. He was doctor honoris causa ofour University, and a honorary freeman of his city Debrecen. In 1980 he was

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International Conference Probability and Statistics with Applications

awarded the Hungarian National Prize (Allami dıj) and in 1999, on his 90thbirthday, the Middle Cross of the Hungarian Republic.

He held leading positions in scientific societies and was honorary mem-ber of diverse institutions. He was an active member of the MathematicalCommittee, as well as the Committee for Computer Science of the HungarianAcademy of Sciences until the end of his life. He was member of the BiometricSociety, the International Statistical Institute, the Bernoulli Society and otherinternational scientific societies. Also he worked as member of the editorialboard of several international journals, in particular of this journal for nearlyfifty years. He was honorary president of the Janos Bolyai Mathematical So-ciety and a honorary member of the Janos Neumann Society of ComputerScience.

He was a very thoughtful man with high moral and ethical standards,highly honored an esteemed by his colleagues and former students.

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Dedicated to the 100th anniversary of the birthday of Béla Gyires

Conf. Place: Auditorium F008-009, Life Science Building Sessions 5, 6 and 9: Auditorium K/3, Chemical Building 3

PROGRAM Tuesday, June 9

Opening Ceremony Chair: A. Pethő

9:00 Pap, Gyula: Béla Gyires (1909-2001)

Remarks: T. Gyires

Coffee break Plenary Session Chair: Gy. Pap

11:00 I. Berkes: Non-central limit theorems for sampling

M. Csörgő: A glimpse of the first 100 year impact of "Student" (1908), The probable error of a mean, Biometrika, 6, 1-25

12-13 Lunch break Session 1. Chair: P. Major

13:00 R. Giuliano: The Rosenblatt coefficient of dependence and the ASCLT for some classes of weakly dependent random sequences

13:20 P. Matula: SLLN for random fields under restrictions on the bivariate dependence structure

13:40 I. Fazekas: An inequality for moments of conditional expectation of random variables and its applications

Coffee break



Tuesday, June 9

Session 2. Chair: M. Csörgő

14:20 Y. Martsynyuk: Invariance Principles and Functional Asymptotic Confidence Intervals for the Slope in Linear Structural and Functional Error- in-Variables Models

14:40 S. Zwanzig: On consistency of R-estimators in errors-in-variables models

Session 3. Chair: J. Sztrik

15:30 Inv.: J. Biró: Loss Ratio Bounds for General Buffered Systems with Regulated Inputs

16:00 L. Bodrog: Statistical analysis of peer-to-peer live streaming traffic

16:20 T. Demián: Fitting of Markov Traffic Models

17:00 Poster Session

Fazekas-Karácsony-Libor: Longest runs in coin tossing. Recursive formulae, asymptotic theorems, computer simulations

T. Tómács: A general method to obtain the rate of convergence in the strong law of large numbers and its applications

J. Túri: Limit theorems for the longest run

Welcome Party at Local Canteen 18:00



Wednesday, June 10

Plenary Session Chair: Gy. Terdik

9:00 T. Subba Rao: Statistical Problems with Spati-Temporal Processes 9:30 J. Bokor: System Identification Using Rational Basis

10:00 T. Gyires: Does The Internet Still Demonstrate Fractal Nature? Coffee break Session 4. Session 5. K/3 Chair: P. Révész Chair: T. Gyires

10:30 P. Kevei: St. Petersburg portfolio games

Z. Gál: Sound synthesis of the QoS mechanisms upon the fully congested IP streams

10:50

M. Barczy: Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

B. Tóth: Hidden Markov Model Based Speaker Dependent and Adaptive Training of Hungarian Text-to-Speech System

11:10 Gy. Pap: Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusions

M. Pinzger: Automatic Web Performance Simulation and Prediction, Based on a Combination of Time Series Analysis and Queuing Network Simulation

11:30 V. Prokaj: Identification of almost unstable Hawkes processes

T. Lukovszki: Phonebook-centric social networks-dealing with similarities

12-13 Lunch break Plenary Session Chair: N. Leonenko

13:00 P. Révész: How short might be the longest run in a dynamical coin tossing sequence

13:30 E. Csáki: Strong limit theorems for a simple random walk on the 2-dimensional comb

Coffee break



Wednesday, June 10

Session 7. Session 6. K/3 Chair: Subba Rao Chair: J. Sztrik

14:20 G. Boshnakov: Non-Gaussian maximum entropy processes

T. Marosits: Flexible Scheduling Discipline for Fixed-Size-Packet Switched Networks

14:40 S. Baran: Parameter estimation in unstable unilateral spatial autoregressive models

Zs. Saffer: Closed form results for BMAP/G/1 vacation model with a class of service disciplines

15:00 M. Rásonyi: On the statistical analysis of quantized Gaussian, AR processes

J. Sztrik: Performance Modeling Tools

15:20

Zs. Karácsony: Asymptotic normality of kernel type regression estimators for random fields

A. Bérczes: Performance Analyzes of a Proxy Cache Server Model with External Users using the Probabilistic Model Checker PRISM

15:40

G. Kusper: Comparing the Performance Modeling Environment MOSEL and the Probabilistic Model Checker PRISM for Modeling and Analyzing Retrial Queuing Systems

16:00 A. Kuki: Experiences with Stochastic Modelling Tools

16:20 G. Fazekas: Software Quality Models: a Probabilistic Approach



Thursday, June 11

Plenary Session Chair: I. Fazekas

9:00 P. Major: The solution of a non-parametric maximum likelihood estimate problem

9:30 N. Leonenko: Statistical inference for Rényi information 10:00 T. Móri: Local properties in scale free random graphs

Coffee break Session 8. Session 9. K/3 Chair: E. Csáki Chair: O. I. Klesov

10:30 Inv.: I. Kátai: Some results and problems in probabilistic number theory

G. Martynov: Cramér-von Mises test for the Weibull and Pareto distributions

11:00 E. Usoltseva: Accelerated Failure Time Model under measurement error

E. Fülöp: Simulations of an HJM type forward interest rate model

11:20

Z. Lagodowski: On almost sure limiting behavior of weighted sums of random fields

B. Nyul: Optimal portfolios and stochastic dominance for some bivariate distributions

11:40 R. Jafri: Comparison of mixtures with non-mixtures

A. Klesov: Rate of Convergence for Certain Optimal Stopping Problems

K. H. Indlekofer: On some methods in the investigation of arithmetical functions

12-13 Lunch break

14:30 Conference excursion, Hortobágy, Places are limited to 45. Bus starts from Chemical Building



Friday, June 12

Plenary Session Chair: T. Móri

9:00 P. Doukhan: Weak dependence, Models and Some Applications

9:30 O. Klesov: Rates of Convergence in Terms of Lévy Distance

10:00 R. Norvaisa: Partial sum processes in p−variation norm Coffee break Session 10. Chair: P. Doukhan

10:30 M. Ispány: Limit theorems for critical inhomogeneous branching processes with immigration

10:50 M. E. Silva: Outliers in INAR(1) models

11:10 Zs. Orlovits: A strong approximation theorem for the estimation of GARCH parameters

11:30 Gy. Terdik: Long-range dependence in third order for non-Gaussian time series

11:50 Closing 12-13 Lunch break

Contents

INVITED TALKS . . . . . . . . . . . . . . . . . . 13Bıro, J.

Loss Ratio Bounds for General Buffered Systems with Regu-lated Inputs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Bokor, J.System Identification Using Rational Basis . . . . . . . . . . . . 15

Csaki, E.Strong limit theorems for a simple random walk on the 2-dimensional comb . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Csorgo, M. and Martsynyuk, V. Y.A glimpse of the first 100 year impact of ”Student” (1908), Theprobable error of a mean, Biometrika, 6, 1-25 . . . . . . . . . . 16

Doukhan, P.Weak dependence, Models and Some Applications . . . . . . . 17

Gyires, T.Does The Internet Still Demonstrate Fractal Nature? . . . . . 18

Katai, I.Some results and problems in probabilistic number theory . . . 19

Klesov, I. O.Rates of Covergence in Terms of Levy Distance . . . . . . . . . 20

Leonenko, N. N.Statistical inference for Renyi information . . . . . . . . . . . . 21

Major, P.The solution of a non-parametric maximum likelihood estimateproblem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Mori, F. T.Local properties in scale free random graphs . . . . . . . . . . 23

Norvaisa, R. and Rackauskas, A.Partial sum processes in p−variation norm . . . . . . . . . . . . 23

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Revesz, P.How short might be the longest run in a dynamical coin tossing

sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Subba Rao, T.

Statistical Problems with Spati-Temporal Processes . . . . . . 25

CONTRIBUTED TALKS . . . . . . . . . . . 26Baran, S. and Pap, Gy.

Parameter estimation in unstable unilateral spatial autoregres-sive models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Barczy, M. and Pap, Gy.Explicit formulas for Laplace transforms of certain functionalsof some time inhomogeneous diffusions . . . . . . . . . . . . . . 27

Berczes, T., Guta, G., Kusper, G., Schreiner, W. andSztrik, J.Comparing the Performance Modeling Environment MOSELand the Probabilistic Model Checker PRISM for Modeling andAnalysing Retrial Queueing Systems . . . . . . . . . . . . . . . 28

Bodrog, L., Horvath, A. and Telek, M.Statistical analysis of peer-to-peer live streaming traffic . . . . 28

Boshnakov, N. G.Non-Gaussian maximum entropy processes . . . . . . . . . . . . 29

Demian, T.Fitting of Markov Traffic Models . . . . . . . . . . . . . . . . . 29

Fazekas, G., Adamko, A. and Arato, M.Software Quality Models: a Probabilistic Approach . . . . . . . 30

Fazekas, I. and Chuprunov, A.An inequality for moments of conditional expectation of ran-dom variables and its applications . . . . . . . . . . . . . . . . 30

Fazekas, I., Karacsony, Zs. and Libor, Zs.Longest runs in coin tossing. Recursive formulae, asymptotictheorems, computer simulations . . . . . . . . . . . . . . . . . 31

Fulop, E. and Pap, Gy.Simulations of an HJM type forward interest rate model . . . 32

Gal, Z. and Terdik, Gy.Sound synthesis of the QoS mechanisms upon the fully con-gested IP streams . . . . . . . . . . . . . . . . . . . . . . . . . 33

Giuliano, R.The Rosenblatt coefficient of dependence and the ASCLT forsome classes of weakly dependent random sequences . . . . . . 34

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Ispany, M.Limit theorems for critical inhomogeneous branching processeswith immigration . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Jafri, R. S.Comparison of mixtures with non-mixtures . . . . . . . . . . . 36

Karacsony, Zs.Asymptotic normality of kernel type regression estimators forrandom fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Kevei, P. and Gyorfi, L.St. Petersburg portfolio games . . . . . . . . . . . . . . . . . . 37

Klesov, O. A.Rate of Convergence for Certain Optimal Stopping Problems . 37

Kuki, A.Experiences with Stochastic Modelling Tools . . . . . . . . . . 38

Kusper, G., Berczes, T., Guta, G., Schreiner, W. andSztrik, J.Performance Analyzes of a Proxy Cache Server Model withExternal Users using the Probabilistic Model Checker PRISM . 39

Lagodowski, A. Z. and Matula, P.On almost sure limiting behavior of weighted sums of randomfields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Lukovszki, T. and Ekler, P.Phonebook-centric social networks-dealing with similarities . . 40

Marosits, T. and Molnar, S.Flexible Scheduling Discipline for Fixed-Size-Packet SwitchedNetworks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Martsynyuk, V. Y.Invariance Principles and Functional Asymptotic ConfidenceIntervals for the Slope in Linear Structural and FunctionalError-in-Variables Models . . . . . . . . . . . . . . . . . . . . . 42

Martynov, V. G.Cramer-von Mises test for the Weibull and Pareto distributions 42

Matula, P.SLLN for random fields under restrictions on the bivariate de-pendence structure . . . . . . . . . . . . . . . . . . . . . . . . 44

Nyul, B.Optimal portfolios and stochastic dominance for some bivariatedistributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Orlovits, Zs. and Gerencser, L.A strong approximation theorem for the estimation of GARCHparameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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Pap, Gy. and Barczy, M.Asymptotic behavior of maximum likelihood estimator for timeinhomogeneous diffusions . . . . . . . . . . . . . . . . . . . . . 46

Pinzger, M.Automatic Web Performance Simulation and Prediction, Basedon a Combination of Time Series Analysis and Queuing Net-work Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Prokaj, V., Torma, B. and Gerencser, L.Identification of almost unstable Hawkes processes . . . . . . . 49

Rasonyi, M.On the statistical analysis of quantized Gaussian, AR processes 50

Rychlik, Z.On the random functional central limit theorems in L2[0, 1]with almost sure convergence . . . . . . . . . . . . . . . . . . . 51

Saffer, Zs. and Telek, M.Closed form results for BMAP/G/1 vacation model with a classof service disciplines . . . . . . . . . . . . . . . . . . . . . . . . 51

Silva, E. M., Barcyz, M., Ispany, M., Pap, Gy. andScotto, M.Outliers in INAR(1) models . . . . . . . . . . . . . . . . . . . 52

Sztrik, J.Performance Modeling Tools . . . . . . . . . . . . . . . . . . . 53

Terdik, Gy.Long-range dependence in third order for non-Gaussian timeseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Tomacs, T.A general method to obtain the rate of convergence in thestrong law of large numbers and its applications . . . . . . . . 54

Toth, B. and Nemeth, G.Hidden Markov Model Based Speaker Dependent and AdaptiveTraining of Hungarian Text-to-Speech System . . . . . . . . . 55

Turi, J.Limit theorems for the longest run . . . . . . . . . . . . . . . . 56

Usoltseva, E.Accelerated Failure Time Model under measurement error . . 56

Zwanzig, S.On consistency of R-estimators in errors-in-variables models . 57

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INVITED TALKS

Non-central limit theorems for sampling

Istvan Berkes

Selection from a finite population is a basic procedure of statistics andlarge sample properties of many classical tests and estimators are closely con-nected with the asymptotic behavior of sampling variables. Typical exam-ples are bootstrap and permutation statistics and their variants. Under mildtechnical conditions, the normed partial sum process Zn(t) of the sampledelements is known to converge weakly to Brownian motion, resp. Brownianbridge according as we select with or without replacement.

The purpose of our talk is to study the behavior of Zn(t) in nonstandardsituations, i.e. when the influence of the large elements of the sample is toostrong for the central limit theorem to hold. We will prove that in this caseZn(t) still converges weakly, but its limit will be nongaussian and it containsrandom parameters. We give a series representation of the limiting processesas sums of independent jump processes. In particular, our results describethe behavior of bootstrap and permutation statistics of samples with infinitevariance. The limit process will be shown to depend sensitively on the tailbehavior of the sampled variables.

Finally, we show how to modify the sampling procedure to avoid thesepathologies.

Loss Ratio Bounds for General Buffered Systems withRegulated Inputs

Jozsef Bıro

During the past few years significant attention has been paid for bufferoverflow probability estimation in case of regulated inputs, both for constantrate servers [1, 2, 3] and general service curve network elements [4, 5, 6]. In [5,6] a long run loss ratio bound has also been presented (see later in Theorem??), which relies on the buffer saturation probability approximations, hence

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we call this indirect bound. Within this paper, conservative, novel direct(definition based) and indirect approximations of the workload loss ratio arepresented for buffered systems with regulated inputs that can be described bya service curve property. If the system is stationary and ergodic the followingdefinition can be used for the stationary long run loss loss ratio1:

LR =E[#of lost bits in a unit time interval]

E[#of bits arriving in a unit time interval]. (1)

It will be shown, that estimating the traffic loss in a direct manner usingthe definition (1) results in closed form bounds. The performance of thesebounds will be compared to that LR bounds published in [5, 6]. For theconstruction of the new bounds only few pieces of information have been usedabout the input traffic (peak rates, upper bound on the mean rate of theaggregated input flows) so it could be directly applied in traffic managementfunctions like call admission control (CAC) as well, without any complexmeasurement or information propagation2.

The novel bounds are derived according to two bounding techniques. Thefirst one [1] based on the decomposition of the investigated network element,into virtual mini-nodes, that process one micro-flow as an input, and has acertain amount of processing capacity, usually a fraction of the entire servercapacity. The summation of the lost traffic in these mini-nodes gives an ap-proximation of the lost traffic within the original system. In the followingsthis approach is referred to as Virtual Node Partitioning (VNP). The otherway to estimate the number of lost packets [2] is named as Busy Period Par-titioning (BPP), since it assigns a union bound for the lost traffic on the timepartition of the maximum possible busy period in which the loss can occur.

References[1] George Kesidis and Takis Konstantopoulos. Worst-case performance ofa buffer with independent shaped arrival processes. IEEE CommunicationLetters, vol. 4, no.1, January 2000.[2] C.-S. Chang, W. Song, and Y. Ming Chiu. On the performance of mul-tiplexing independent regulated inputs. In proceedings of Sigmetrics, pages:184-193, May 2001.

1Without loss of generality we consider a bit-processing system, since it can be shown,that the result can be converted for systems with higher granularity (cells, packets)

2The loss ratio bounds in [5, 6] require the exact average aggregate arrival intensity ofthe inputs (denoted by ρ in that paper and also in this work) which may need measurementin an implementation. The novel bounds to be presented in this paper make possible torelax this requirement by using a conservative upper bound on the mean rate, which doesnot need measurement.

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[3] F. M. Guillemin, N. Likhanov, R. R. Mazumdar, C. Rosenberg, and YuYing. Buffer overflow bounds for multiplexed regulated traffic streams. InITC-18, Munich, Germany, Sept 2003.[4] Milan Vojnovic and Jean-Yves Le Boudec. Bounds for independent regu-lated inputs multiplexed in a service curve network element. IEEE Transac-tions on Communications 51(5): 735-740, May 2003.[5] Milan Vojnovic and Jean-Yves Le Boudec. Stochastic analysis of someexpedited forwarding networks. In proceedings of INFOCOM Vol. 2 pages:1004-1013, June 2002.[6] M. Vojnovic and J. Y. Le Boudec. Stochastic analysis of some expeditedforwarding networks. Technical Report DSC/2001/039, EPFL-DI-ICA, July2001.

System Identification Using Rational Basis traffic

Jozsef Bokor

The role of system identification (ID) is to generate dynamic models frommeasured data. The selection of a suitable model set to represent a partic-ular dynamic phenomena is one of the most important step. When dealingwith linear time invariant systems, one possible choice is the family of linearlyparametrized models leading to impulse response or infinite MA representa-tions. In sampled time domain ID the classical approach is to choose a setof basis functions usually generated by the canonic delay operator. This con-cept can be generalized if using rational functions of the delay like in case ofLaguerre and Kautz bases.

This talk will discuss the parameter estimation problem in general rationalbases related to the Blaschke - functions. Variance and bias of these estimatedwill be approximated using results from Szego. These results were broughtinto the author’s attention by Professor Bela Gyires.

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Strong limit theorems for a simple random walkon the 2-dimensional comb

Endre Csaki

We study path behaviour of a simple random walk on the 2-dimensionalcomb lattice C2 that is obtained from Z2 by removing all horizontal edges offthe x-axis. Let C(n) = (C1(n), C2(n)), n = 0, 1, 2, . . . be a simple randomwalk on C2. The transition probability is given as

P(C(n+ 1) = (x, y ± 1) | C(n) = (x, y)) = 1/2, if y 6= 0,P(C(n+ 1) = (x± 1, 0) | C(n) = (x, 0)) = 1/4,P(C(n+ 1) = (x,±1) | C(n) = (x, 0)) = 1/4.

Bertacchi (2006) proved a weak convergence result(C1(nt)n1/4

,C2(nt)n1/2

; t ≥ 0)Law→ (W1(η2(0, t)),W2(t); t ≥ 0), n→∞,

where W1, W2 are two independent Wiener processes (Brownian motions)and η2(0, t) is the local time process of W2 at zero, We prove a strong ap-proximation analogue of this result, and obtain further strong limit theorems,like joint Strassen type law of the iterated logarithm, as well as Hirsch typebehaviour of the two components.

This presentation will mainly be based on joint work with M. Csorgo, A.Foldes and P. Revesz, available as Technical Report Series of the Laboratoryfor Research in Statistics and Probability, No. 442 - February 2009, CarletonUniversity - University of Ottawa, and arXiv:0902.4369.

A glimpse of the first 100 year impact of”Student” (1908), The probable error of a mean,

Biometrika, 6, 1-25

Miklos Csorgo and Yuliya V. Martsynyuk

This talk is in celebration of the centennial of one of the most famousseminal contributions in 20th century statistics, Student (1908).

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After giving a snapshot of the original Student (1908) contributions, wewill review the modern large-sample theory and its ramifications for the Stu-dent statistic and processes, including some of the speakers’ most recent worksin this area. In particular, we will take a detailed look at the paper by Yu. V.Martsynyuk, Functional asymptotic confidence intervals for a common meanof independent random variables, Electronic Journal of Statistics, 3 (2009),25-40, that deals with functional central limit theorems for Studentized par-tial sums of independent random variables with a common mean that eithersatisfy Lindeberg’s condition, or are symmetric around the mean. We thenconclude by presenting best possible weighted sup-norm approximations, aswell as weighted Lp-approximations, in probability, for Student processes inD[0,1], as in the paper M. Csorgo, B. Szyszkowicz and Q. Wang, Acta Math.Hungar., 121 (4) (2008), 307-332.

Weak dependence, Models and Some Applications

Paul Doukhan

The talk will present the basic features of weak dependence defined in Doukhan& Louhichi (1999). A book on the subject coauthored with Dedecker, Lang,Leon, Louhichi and Prieur appeared as LNS 190 (Springer) in 2007.Our main aim is to propose several new models of time series and randomfields for which this theory applies beyond mixing. Nonlinear, nonMarkovstationary models will be proposed.We shall derive from a Lindeberg type theorem some applications to resam-pling and functional estimation. A fast evocation of applications will be pro-vided. A special attention will be given to applications reated with extremevalues theory, namely its translation in terms of weak dependence assump-tions as well as the way subsampling methods apply to estimate confidenceintervals, or quantiles of the limit distributions.

References[1.] Dedecker, J. and Doukhan, P., Prieur, C., Louhichi, S., Dedecker, J.,Leon, J. R. (2007) Weak dependence: models, theory and applications (350pages) Lecture Notes in Statistics 190, Springer-Verlag.[2.] Doukhan, P. and Louhichi, S. (1999) A new weak dependence conditionand applications to moment inequalities. Stoch. Proc. Appl. 84, 313-342.[3.] Doukhan, P., Robert, C. Y. (2009) The condition D(un) under weakdependence with applications to extreme value theory.[4.] Doukhan, P., Prohl, S. (2009) Subsampling extreme statistics under weakdependence conditions with applications to LARCH type models.

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Does The Internet Still Demonstrate Fractal Nature?

Tibor Gyires and Gyorgy Terdik

Measurements of local and wide-area network traffic in the 90’s establishedthe relation between burstiness and self-similarity of network traffic. Severalpapers demonstrated that the widely used Poisson based models could notbe applied for the past decades network traffic. If the traffic had been aPoisson process, the traffic’s burst lengths would have been smoothed byaveraging over a long time scale contradicting with the observations of the pastdecade’s traffic characteristics. Poisson models were abandoned as unsuitablecharacterizations of network traffic.

Recent papers have questioned the direct applicability of these results innetworks of the new century. Some authors of these papers demand the revi-sion of previous assumptions on the Poisson traffic models. They argue thatas newer and newer network technologies are implemented and the amountof Internet traffic grows exponentially, the burstiness of network traffic mightcancel out due to the huge number of aggregated traffic flows. Some resultsare based on analyses of high-speed Internet backbone links and other traffictraces. We analyzed the same traffic traces and applied novel methods tocharacterize them in terms of packet interarrival time. We demonstrate thatthe series of interarrival times in the 2003 traces is still close to a selfsimi-lar process. Since then, new traffic traces have been made public, includingones captured from OC-192 links of the Internet backbone in 2008. We alsocompare the 2008 traffic traces with the ones captured in 2003 and applyour analytical methods to illustrate the tendency of Internet traffic bursti-ness in recent years. We found that the burstiness of the interarrival timesdecreased significantly compared to earlier traces. In addition, we presentspecial techniques applied in time-stamping of the captured packet traces.

We briefly describe the problems associated with the network measurementmethods designed to provide very high quality packet time-stamps in high-speed networks.

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Some results and problems inprobabilistic number theory

Imre Katai

A function f : N→ R is said to be additive, if f(m,n) = f(m)+f(n) holdsfor all coprime pairs of m,n ∈ N. A function g : N → C is multiplicative, ifg(m,n) = g(m) · g(n) holds for all coprime pairs of m,n.

Let q ≥ 2 be an integer, Aq = 0, ...q − 1. Then n ∈ N can be expandedin the form:

n = ε0(n) + ε1(n)q + ...+ εk(n)gk,

εj(n) ∈ Aq, εk(n) 6= 0

uniquely. The function f : N0 → R is said to be q−additive, if f(0) = 0, and

f(n) =k∑j=0

f(εj(n)gj) (n ∈ N)

The function g : N0 → C is q−multiplicative, if g(0) = 1, and

g(n) =k∏j=0

g(εj(n)gj) (n ∈ N)

The main aim of probabilistic number theory is to give necessary and suffi-cient conditions for additive (q−additive) functions to have limit distributionafter suitable centralization or normalization. These questions are more dif-ficult, if we consider the distributions on some arithmetically characterizedsubsets of the integers: e.g. on the set of shifted primes, on the set of poly-nomial values, on the set of integers having exactly k prime divisors. Themean-values of multiplicative (q−multiplicative) functions over some subsetsof integers are also considered.

19


Rates of Covergence in Terms of Levy Distance

Oleg I. Klesov

We discuss the following (and some other) results.Theorem. Let F be a distribution function. Denote by L the Levy distancebetween F and the standard Gaussian law Φ. Let p > 0. Assume that∫ ∞

−∞|x|p dF <∞.

Put

λ =∣∣∣∣∫ ∞−∞|x|p dF −

∫ ∞−∞|x|p dΦ

∣∣∣∣ . (1)

Let L0 < 1 be fixed. Assume that

L ≤ L0.

Then there exists a universal constant c (depending on p and L0) such that

|F (x)− Φ(x)| ≤λ+ cL

(ln 1

L

)p/21 + |x|p

Theorem. Let Fn be a sequence of distribution functions and let Ln denotethe Levy distance between Fn and Φ. Assume that

lim supn→∞

Ln < 1.

Let p > 0 andlim supn→∞

λn <∞

where λn is defined for Fn similarly to (1). Let r > 0 be fixed and let afunction f be positive. If ∫ ∞

−∞

f(x)(1 + |x|p)r

dx <∞,

thenlim supn→∞

∫ ∞−∞

f(x)|Fn(x)− Φ(x)|r dx <∞.

Moreover ifFn

w→ Φ,

20


thenlimn→∞

∫ ∞−∞

f(x)|Fn(x)− Φ(x)|r dx = 0. (2)

Equality (2) for f(x) ≡ 1 is called the global version of the central limittheorem.

Statistical inference for Renyi information

Nikolai N. Leonenko

We present a new class of estimators for the Renyi information (see Renyi(1961)) of multi-dimensional probability density, based on the k-th nearestdistances in a sample of i.i.d. vectors (see Leonenko, Pronzato and Savani(2008)). The method can be extended to the estimation of the statisticaldistances between two distributions using one i.i.d. sample from each. Anapplications of different entropies (ε-entropy and quadratic Renyi entropy, seeLeonenko and Seleznev (2009)) are also discussed. The paper by Baryshnikovet al. (2009) contains general statistical properties (including asymptoticnormality) for functionals which contain entropies as partial case, but theyhold only for a densities with bounded support. For the other approachesbased on Voronoi tessellations, see, Jimenez and Yukich (2002), Learned-Miller and Fisher (2003). In fact, one can consider Voronoi or Dirichlet cellsas a multidimensional ”geometrical” generalization of the one-dimensionalspacings, while the nearest neighbor balls as probabilistic counterpart of one-dimensional spacings, see Ranneby et al. (2005) for a discussion. ProfessorBela Gyires made a major contribution the theory of ordered statistics andspacings (see Gyires [3]-[5]). Also Professor Bela Gyires published a paperabout Alfred Renyi (see Gyires (1970)).

References[1] Baryshnikov Yu, Penrose, M. and Yukich, J.E. (2009) Gaussian limits forgeneralized spacings, Annals of Applied Probability, in press.[2] Gyires, B. (1971) Alfred Renyi (1920–1970). Publ. Math. Debrecen 17 ,1–17 .[3] Gyires, B.(1975) Linear order statistics in the case of samples with nonin-dependent elements. Publ. Math. Debrecen 22, no. 1-2, 4763[4] Gyires, B. (1977) Normal limit-distributed linear order statistics, SankhyaSer.A 39, no. 1, 11–20.[5] Gyires, B. (1982) Doubly ordered linear rank statistics, Acta Math. Acad.

21


Sci. Hungar. 40, no. 1-2, 5563[6] Jimenez, R. and J.E. Yukich J.E. (2002) Asymptotics for statistical dis-tances based on Voronoi tessellations. J.Theoret.Probab. 15, 503–541.[7] Leonenko, Nikolai; Pronzato, Luc; Savani, Vippal A class of Renyi infor-mation estimators for multidimensional densities. Ann. Statist. 36 (2008),no. 5, 21532182[8] Leonenko, N. N., Seleznev, O. (2009) Statistical inference for ε-entropyand quadratic Renyi netropy, J. Multivariate Analysis, submitted[9] Learned-Miller, E. and Fisher, J. (2003) ICA using spacing estimation ofentropy, J. Machine Learning Research 4, 1271–1295.[10] B. Ranneby, B., Jammalamadaka, R. and Teterukovskiy, S.A. (2005) Themaximum spacing estimation for multivariate observations, J. Stat. Plann.Infer., 129, 427–446.[11] Renyi, A (1961) On measures of entropy and information. In 4th BerkleySymp. Math. Statist. Prob., vol. I, pp. 547–561, Univ. Calif. Press,Berkeley, 1961.

The solution of a non-parametric maximum likelihoodestimate problem

Peter Major

The error of maximum likelihood estimates satisfy a central limit theoremunder very general conditions, and this result is one of the main reason of theimportance of this method. The question arises whether a similar method canbe given for the solution of non-parametric estimation problem, and a similargood estimate can be proved for their error.

In this talk I shall give a positive answer in a special case, in the solutionof the socalled Kaplan-Meyer method for the estimation of a distributionfunction with the help of censored data. The proof is similar to the case ofmaximum likelihood estimates, but the linearization argument with the help ofa Taylor expansion argument has to be replaced by a different argument. Herea new problem appears which is interesting in itself. We need good estimateson multiple integrals with respect to a normalized empirical distribution.

22


Local properties in scale free random graphs1

Tamas F. Mori

Real life networks are often very large and have complex structures. More-over, they usually possess the so called scale free property, that is, their degreedistributions decay at polynomial rates: the ratio of vertices with degree d isapproximately C ·d−γ for large d. Their size and complexity justifies the usingof stochastic models. In the last decades several models of evolving randomgraph processes have been invented and studied. For model fitting it is neces-sary to derive reasonably good estimations for important parameters, such asthe characteristic exponent γ. The problem is that the observer only knowsthe graph locally: just a small subset of the nodes can be accessed. That maymade the global estimators biased. For example, in many scale free modelsit was observed that the characteristic exponent changed when attention wasrestricted to a set of selected vertices that were close to the initial configu-ration. In the talk this phenomenon is investigated. Under some conditionsimposed on the original graph and the selection procedure we point out howthe size of the selected set influences the observed characteristic exponent.

Partial sum processes in p−variation norm

Rimas Norvaisa and Alfredas Rackauskas

During the talk we discuss some old and new results concerning asymptoticbehaviour of a partial sum process in the p−variation norm. To fix notationlet X1, X2, ... be a sequence of real-valued random variables. For each integern ≥ 1 and real t ∈ [0, 1], let

Sn(t) :=bntc∑i=1

=n∑i=1

Xi1[0, t](i

n

),

where bxc is the integer part of x and 1B is the indicator function of a setB. Then the partial sum process is the sequence of nth partial sum processes

1Parts of this research are joint work with Agnes Backhausz. It has been supported bythe Hungarian Scientiffic Research Fund (OTKA) Grant K67961.

23


Sn = Sn(t) : t ∈ [0, 1], n ≥ 1. For a function f : [0, 1] → R and a numberp ∈ (0,∞), the p−variation of f is

vp(f) := sup

m∑i=1

|f(ti)− f(ti−1)|p : 0 = t0 < t1 < ... < tm = 1, m ∈ N+

.

If vp(f) < ∞ then f has bounded p−variation and the set of all such func-tions is denoted by Wp[0, 1]. For each f ∈ Wp[0, 1] and 1 ≤ p < ∞, let‖f‖[p] := ‖f‖sup + vp(f)1/p, where ‖f‖sup := sup |f(x)| : x ∈ [0, 1] . The setWp[0, 1] is a Banach space with the norm ‖·‖[p].

The p−variation for a sample function of the nth partial sum Sn is

vp(Sn) = max

m∑j=1

∣∣∣∣∣∣kj∑

i=kj−1+1

Xi

∣∣∣∣∣∣p

: 0 = k0 < ... < km = n, 1 ≤ m ≤ n

,

and so it is bounded for any 0 < p <∞.

We recall main results concerning partial sum processes in p−variation.Suppose that X1, X2, ... are independent mean zero random variables withpth moment finite for some p ∈ [1, 2). J. Bretagnolle (1972) proved that thereexists a constant Cp depending on p only and such that(

n∑i=1

E |Xi|p ≤

)Evp(Sn) ≤ Cp

n∑i=1

E |Xi|p .

Suppose that random variables X1, X2, ... are independent identically dis-tributed, EX1 = 0 and EX2

1 = 1. Let Lx := max 1, log x , x > 0. J. Qian(1998) proved that v2(Sn) = OPr(nLLn) as n → ∞. Also OPr(nLLn) cannot be replaced by oPr(nLLn) provided E |X1|2+ε for some ε > 0.

Our contribution is the following result. Let 2 < p < 1 and let W =W (t) : t ∈ [0, 1] be a Wiener process. The convergence

n−1/2Sn ⇒W in law in Wp[0, 1],

as n→∞ holds if and only if EX1 = 0 and EX21 = 1.

References[1] J. Bretagnolle, p−variation de fonctions alatoires. 2ieme partie: Processusa accroissements independants. Lect. Notes Math., 258, 64-71 (1972).[2] R. Norvaisa and A. Rackauskas, Convergence in law of partial sum pro-cesses in p−variation norm. Lithuanian Math. J., 48, 212-227 (2008).[3] J. Qian, The p−variation of partial sum processes and the empirical pro-cess. Ann. Probab., 26, 1370-1383 (1998).

24


How short might be the longest run in adynamical coin tossing sequence

Pal Revesz

Let X1, X2, . . . denote i.i.d. random bits each taking the values 1 and 0with respective probabilities 1/2 and 1/2. A well-known theorem of Erdosand Renyi ([2]) describes the limit distribution of the length of the longestcontiguous run of ones in X1, X2, . . . , Xn as n → ∞. Benjamini et al. ([1]Theorem 1.4) demonstrated the existence of unusual times, provided that thebits undergo equilibrium dynamics in time. In fact they prove that the dy-namics produces much longer runs than the original model. In the presentpaper we study the length of the shortest run in the presence of the dynamics.

Keywords: Dynamical limit theorems, exceptional times, run tests, randomwalk, strong theorems

AMS 2000 subject classif ications: 60J05, 60F15, 60F17 .

Statistical Problems with Spati-Temporal Processes

Tata Subba Rao

We briefly describe some approaches to test for nonlinear dependence ofspatial, spatio temporal processes. Also frequency domain approach to pre-diction which has not been considered before. The ideas are exploratory andwe are in the process of testing on real data sets.

25


CONTRIBUTED TALKS

Parameter estimation in unstable unilateralspatial autoregressive models 1

Sandor Baran and Gyula Pap

We consider spatial autoregressive processes of the form

Xk,` = αXk−1,` + βXk,`−1 + γXk−1,`−1 + εk,`, (1)

where the independent innovations εk,` have zero mean and unit variance.The model is stable if |α| < 1, |β| < 1 and |γ| < 1, |1 + α2 − β2 − γ2| >2|α + βγ| and 1 − α2 > |α + βγ| and unstable on the boundary of thisdomain (Basu and Reinsel, 1993). We investigate the asymptotic propertiesof the least squares estimator (LSE) of the parameters in the unstable case.

First we consider the special case γ = 0 and show that the LSE of (α, β)is asymptotically normal (Baran et al., 2007) and the rate of convergenceis n−3/2. Further we provide some new results concerning the parameterestimation in the general model (1).

References[1] Basu, S. and Reinsel, G. C. (1993) Properties of the spatial unilateralfirst-order ARMA model, Adv. Appl. Prob., 25, 631-648.[2] Baran, S., Pap, G., Zuijlen, M. v. (2007) Asymptotic inference for unitroots in spatial triangular autoregression. Acta Appl. Math., 96, 17-42.

References[1] Barczy, Matyas; Pap, Gyula. Asymptotic behavior of maximum likelihoodestimator for time inhomogeneous diffusion processes. (2008).Arxiv: 0810.2688, URL: http://arxiv.org/abs/0810.2688[2] Bishwal, Jaya P. N.. Parameter estimation in stochastic differential equa-tions. Springer-Verlag, Berlin-Heidelberg, 2008. ISBN: 978-3-540-74447-4[3] Kutoyants, Yury A.. Statistical Inference for Ergodic Diffusion Processes.Springer-Verlag, London, 2004. ISBN: 1-85233-759-1[4] Luschgy, Harald. Local asymptotic mixed normality for semimartingaleexperiments. Probability Theory and Related Fields, 92 (1992), 151–176.

1Research is partially supported by the Hungarian National Science Foundation OTKAunder Grant No. T079128/2009.

26


Explicit formulas for Laplace transforms of certainfunctionals of some time inhomogeneous diffusions

Matyas Barczy and Gyula Pap

We consider a process (X(α)t )t∈[0,T ) given by the SDE dX(α)

t = αb(t)X(α)t dt+

σ(t) dBt, t ∈ [0, T ), with initial condition X(α)0 = 0, where T ∈ (0,∞],

α ∈ R, (Bt)t∈[0,T ) is a standard Wiener process, b : [0, T ) → R \ 0 andσ : [0, T )→ (0,∞) are continuously differentiable functions. Assuming

ddt

(b(t)σ(t)2

)= −2K

b(t)2

σ(t)2, t ∈ [0, T ),

with some K ∈ R, we derive an explicit formula for the joint Laplacetransform of ∫ t

0

b(s)2

σ(s)2(X(α)

s )2 ds and (X(α)t )2

for all t ∈ [0, T ) and for all α ∈ R. Our motivation is that the MLE of αcan be expressed in terms of these random variables.

As an application, we prove asymptotic normality of the MLE αt of αas t ↑ T for sign(α −K) = sign(K), K 6= 0. We also show that in case ofα = K, K 6= 0,

√IK(t) (αt −K) L= − sign(K)√

2

∫ 1

0Ws dWs∫ 1

0(Ws)2 ds

, ∀ t ∈ (0, T ),

where IK(t) denotes the Fisher information for α contained in the observa-tion (X(K)

s )s∈[0, t], (Ws)s∈[0,1] is a standard Wiener process and L= denotesequality in distribution. As an example, for all α ∈ R and T ∈ (0,∞), westudy the process (X(α)

t )t∈[0,T ) given by the SDEdX(α)

t = − αT−tX

(α)t dt+ dBt, t ∈ [0, T ),

X(α)0 = 0.

In case of α > 0, this process is known as an α-Wiener bridge (see [3]), andin case of α = 1, this is the usual Wiener bridge. We prove that for allα, β ∈ R, α 6= β, the probability measures induced by the processes X(α)

and X(β) are singular on (C[0, T ), B(C[0, T ))). Further, we investigateregularity properties of X

(α)t as t ↑ T .

27


References[1] M. Barczy and G. Pap, Explicit formulas for Laplace transforms of cer-tain functionals of some time inhomogeneous diffusions. arXiv:0810.2930v1URL: http://arxiv.org/abs/0810.2930 (2008).[2] M. Barczy and G. Pap, α-Wiener bridges: singularity of induced mea-sures and sample path properties. To appear in Stochastic Analysis and Appli-cations. Available: arXiv:0810.3070, URL: http://arxiv.org/abs/0810.3070[3] R. Mansuy, On a one-parameter generalization of the Brownian bridgeand associated quadratic functionals. Journal of Theoretical Probability 17(4),1021–1029 (2004).

Comparing the Performance Modeling EnvironmentMOSEL and the Probabilistic Model Checker PRISMfor Modeling and Analysing Retrial Queueing Systems

Tamas Berczes, Gabor Guta, Gabor Kusper,

Wolfgang Schreiner and Janos Sztrik

We describe the results of analyzing the performance model of a retrialqueueing system with the probabilistic model checker PRISM. The systemhas been previously analyzed with the help of the performance modeling en-vironment MOSEL; we are able to accurately reproduce the results reportedin literature. Furthermore, we compare PRISM and MOSEL with respectto their modeling languages and ways of specifying performance queries andbenchmark the executions of the tools.

Statistical analysis of peer-to-peerlive streaming traffic

Levente Bodrog, Akos Horvath and Miklos Telek

Effective network dimensioning and usage requires an accurate analysis ofthe network traffic components. A recently emerging network traffic compo-nent is the peer-to-peer (P2P) live streaming (TV) traffic. The paper analyseits stochastic behaviour based on the data sets collected at a recent measure-ment campaign. Apart of the commonly applied (short range) statistical testswe devote special attention to the long range behaviour of the traffic process.

28


Non-Gaussian maximum entropy processes

Georgi N. Boshnakov

We solve the maximum entropy problem for autocovariances given overa general subset of N. We also consider and solve a more general problemwhere prediction coefficients and prediction error variances are given instead ofcovariances. We introduce two notions about maximum entropy in time seriescontext, obtain non-Gaussian solutions, and discuss some misconceptions inthe literature.

Fitting of Markov Traffic Models

Tamas Demian

Authors try to find appropriate similarity transformation that could con-vert a given ME (Matrix Exponential) representation to a more favourablePH (Phase Type) representation. Authors give necessary conditions for theexistence of such a representation and give methods for the search. They giveconjectures on necessary and sufficient conditions too. PH distribution is thedistribution of the time until absorption into the absorbent state in a Markovchain. If the arrival and service time distributions are PH distributions ina queuing system, we can use simple linear algebraic methods to derive themost important features or to perform simulation. Robust methods exist thatcan approximate any distribution with a ME distribution (with respect to agiven measure and matrix order), but the PH transformation have not beensufficiently examined yet. This transformation is the object of the currentpresentation.

Keywords: Markov chain, PH distribution, ME distribution, traffic models,cyclic matrix

29


Software Quality Models: a Probabilistic Approach

Attila Adamko, Matyas Arato and Gabor Fazekas

In this paper we would like to give an overview about software quality metrics.In particular, metrics that are used to measure change in a project and/or its entitiestogether with the measuring of fine-grained changes. Measuring change is importantfor several reasons: it helps understand the direction that the software product takes,or it can help evaluate the work done by different development teams and metricscan also help developers to improve their programming and design practices andalso help researchers to understand how software evolves.

In order to define metrics we need information about the topically significantpart. What significant is always depends on the data available and the measuringobjectives. We could measure how much a software system has been modified be-tween two versions of it. This modification can be fine grained (measured by lines ofcode), or more coarse grained, like the difference between two releases of the system.The expectation is that this comparison will tell us something about how the sys-tem has evolved during the observed period. The Cyclomatic complexity measureis commonly used in this manner as a way to detect erosion in a software system.Because it is a high-level notation made up of many different attributes, there cannever be a single measure of software complexity. More of these are restricted tocode. When we would like to measure the users view of a systems functionality wecould use Albrechts function points (FP) metric.

The major benefit of FPs is that they are not restricted to code. In fact,they are computed from a detailed system specification. Anyway, because of thehuge amount of data and their uncertain character a probabilistic model seems tobe adequate.

An inequality for moments of conditional expectation ofrandom variables and its applications

Alexey Chuprunov and Istvan Fazekas

Consider the probability measure P and the conditional probability measurePA with respect to the fixed event A. Let EA denote the expectation with respectto PA. We prove the following inequality for centered moments of random variables

EA∣∣∣S − EAS

∣∣∣p ≤ 4pE |S − ES|p

P(A)

for 1 ≤ p <∞. This inequality is applied to the generalized allocation scheme. Letη1, ..., ηN be nonnegative integer-valued random variables. If there exist independent

30


random variables ξ1, ..., ξN such that the joint distribution of η1, ..., ηN admits therepresentation

Pη1 = k1, ..., ηN = kN = Pξ1 = k1, ..., ξN = kN |ξ1 + ...+ ξN = n,

where k1, ..., kN are arbitrary nonnegative integers, we say that η1, ..., ηN represent ageneralized allocation scheme with parameters n and N , and independent randomvariables ξ1, ..., ξN . We assume that the distribution of ξi is

Pξi = k =bkθ

k

k!B(θ), k = 0, 1, 2, ...

where θ > 0. Here b0, b1, ... is a sequence of non-negative numbers with b0 > 0, b1 > 0and we assume that the convergence radius of the series

B(θ) =

∞∑k=0

bkθk

k!

is positive. This scheme is widely studied, see [1], [2].

References[1] Kolchin V.F. Random Graphs. Cambridge University Press, Cambridge, 1999.[2] Kolchin, A. V. Limit theorems for a generalized allocation scheme. (Russian)Diskret. Mat. 15 (2003), no. 4, 148157; translation in Discrete Math. Appl. 13(2003), no. 6, 627636.

Longest runs in coin tossing.Recursive formulae, asymptotic theorems,

computer simulations

Istvan Fazekas, Zsolt Karacsony and Jozsefne Libor

The coin tossing experiment is studied. The length of the longest head runcan be studied by asymptotic theorems ([2], [3]), by recursive formulae ([5], [4]) orby computer simulations ([1]). The aim of the paper is to compare numerically theasymptotic results, the recursive formulae, and the simulation results. Moreover, weconsider also the longest run (i.e. the longest pure heads or pure tails). We comparethe distribution of the longest head run and that of the longest run. We considerboth fair and biased coins.

References[1.] Binswanger, K.; Embrechts, P. Longest runs in coin tossing. Insurance Math.

31


Econom. 15 (1994), no. 2-3, 139–149.[2.] Erdos, P.; Revesz, P. On the length of the longest head-run. Topics in infor-mation theory (Second Colloq., Keszthely, 1975), pp. 219–228. Colloq. Math. Soc.Janos Bolyai, Vol. 16, North-Holland, Amsterdam, 1977.[3.] Foldes, A. On the limit distribution of the longest head run. (Hungarian) Mat.Lapok 26 (1975), no. 1-2, 105–116 (1977).[4.] Kopociski, B. On the distribution of the longest success-run in Bernoulli trials.Mat. Stos. 34 (1991), 3–13.[5.] Schilling, Mark F. The longest run of heads. College Math. J. 21 (1990), no.3, 196–207.

Simulations of an HJM typeforward interest rate model

Erika Fulop and Gyula Pap

We consider a discrete time Heath-Jarrow-Morton (HJM) type forward interest ratemodel driven by a spatial autoregression process. Let ηi,j : i, j ∈ Z+ be i.i.d. stan-dard normal random variables on a probability space (Ω,F , P ), where Z+ denotesthe set of nonnegative integers. Let ρ ∈ R be the autoregression coefficient. Letfk,` denote the forward interest rate at time k with time to maturity date `(k, ` ∈ Z+). We assume, that the initial values f0,j are known at time 0. Next,we suppose that the market is arbitrage free. Then the forward rates are given bythe following equations:

fk,` = fk−1,`+1 + ρ(fk,`−1 − fk−1,`) + ηk,` +1

2

2∑i=0

ρi, (k, ` ∈ N)

(see [4]).Our aim is to simulate this process in stable cases, when |%| < 1, and also

in unstable cases, when |%| = 1. We analyse the numerical problems related tomaximum likelihood estimation of the autoregression parameter %. Finally westudy the asymptotic behavior of this maximum likelihood estimator.

References[1.] Fulop, E. and Pap, G. (2007), Asymptotically optimal tests for a discrete timerandom field HJM type interest rate model, Acta Sci. Math. , 73(3-4), 637–661.[2.] Fulop, E. and Pap, G. (2008), Note on strong consistency of maximum likeli-hood estimators for dependent observations, Proc. of the 7th International Confer-ence on Applied Informatics (Eger, 2007), Volume 1, pp. 223–228.[3.] Fulop, E. and Pap, G. (2009), Forward interest rate curves in discrete time

32


settings driven by random fields, Lithuanian Math. J., 49(1), 5–25.[4.] Gall, J. (2008), Some Problems in Discrete Time Financial Market Models,Ph.D. dissertation, University of Debrecen, Hungary.[5.] Gall, J., Pap, G. and Peeters, W. (2007), Random field forward interestrate models, market price of risk and their statistics, Ann. Univ. Ferrara Sez. VIISci. Mat. 53, 233–242.

Sound synthesis of the QoS mechanisms upon the fullycongested IP streams

Zoltan Gal and Gyorgy Terdik

Expectations of modern communication switched networks are accentuatedtoday by the real time transmission demand of data, voice (VoIP) and video oncommon network infrastructure. One of the realization possibilities of the multi-media network services is the assurance of data transfer at higher channel ratesthan the necessary capacity for local applications. Coverage of this condition inLAN/MAN environment can be guaranteed relative simply without strict networkresource design. This aspect is exploited by some traders, which produce low pricedcommunication devices with high speed interfaces (1-10 Gigabit/sec) but low switch-ing intelligence. In the other side lot of desktops and workstations with 1 Gigabit/secinterfaces running multimedia network applications (like HD videoconference, etc.)consume more and more network bandwidth, which makes necessary QoS serviceguarantees for intermediate nodes. This implies enough intelligence integrated inswitches to apply DiffServ/IntServ/RSVP mechanisms in optimum conditions toprovide satisfactory network resources for real time traffic.

In both cases of the applied enterprise infrastructure development strategiesconsiderations regarding resource utilization of the different bit flows are neededtoday. Evaluation of statistical aspects (LRD - Long Range Dependency, Self Sim-ilarity, Fractals, etc.) of PDU transmission processes and the Corvil bandwidth ofdifferent data flows give sophisticated metric characteristics.

Traits of sampled best effort data flows will be presented in the paper, whereQoS controlled VoIP network traffics and TCP/UDP based data traffics are trans-mitted in fully congested Ethernet trunk link. Characterization of different trafficsare based on wavelet analysis and highlight of a new evaluation method for wiredswitched traffics, called ON/(ON+OFF) transformation will be submitted. Becauseboth the well known melodic music and the classical switched network traffic haslong range dependent characteristics the effects of QoS controlled IP packet flow onthe simultaneous best effort data traffic will be emphasized with live stereo songdemonstration as well. Based on these considerations practical proposals regardingreal time supervision of the switched IP LAN/MANs task will be drawn for IP net-work service providers.

33


Keywords: VoIP, QoS, DiffServ/IntServ/RSVP, LRD, self similarity, fractal, Corvilbandwidth, entropy

References[1] Leland, W. E., Taqqu, M. S., Willinger, W., Wilson, D. V. (1994), On the self-similar nature of Ethernet traffic (extended version), IEEE/ACM Transactions onNetworking (TON), Volume 2, Issue 1, February 1994, ISSN:1063-6692.[2] Park, K., Willinger, W. Eds. (2000). Self-Similar Network Traffic and Perfor-mance Evaluation. Wiley-Interscience, New York. ISBN: 978-0-471-31974-0.[3] E. Igloi; Gy. Terdik, (2003), Superposition of Diffusions with Linear Generatorand its Multifractal Limit Process, ESAIM: Probability and Statistics, vol. 7, pp.23-88.[4] Patrice Abry (2001), Lois Dechelle, Multiresolutions et Ondelettes,HabilitationTravaux de Recherche, Universit Claude Bernard Lyon, Mars 2001.[5] Brillinger, D. R. Irizarry, R.A. An investigation of the second-and higher-orderspectra of music Signal Processing, Elsevier, Signal Processing 65 (1998) 161-179[6] Rafael A Irizarry Loca l Harmonic Estimation in Musical Sound Signals ...Journal of the American Statistical Association. June 1, 2001, 96(454): 357-367.doi:10.1198/016214501753168082..

The Rosenblatt coefficient of dependence and theASCLT for some classes of weakly dependent random

sequences

Rita Giuliano

The Almost Sure Central Limit Theorem (ASCLT) is a classical result inthe asymptotic theory of random sequences. It has been originally proved for asequence of i.i.d. random variables (Xn)n≥1 successively it has been extended tomore general sequences: the typical framework is that of a stationary sequence,weakly dependent in some sense. For instance associated, α−mixing and ρ−mixingsequences; or sequences dependent in the sense of Doukhan and Louhichi.

The aim of this talk is to discuss an ASCLT for sequences such that theirpartial sums Sn (properly normalized) have a ”good” speed of convergence to theGaussian law.

As an example, we present the case of stationary α−mixing sequences; how-ever, our point of view allows to study also the case of some non-stationary se-quences: we shall consider sequences satisfying another condition of dependence,introduced by E. Rio.

We prove the ASCLT by means of a new bound for the Rosenblatt coefficientof normalized partial sums, obtained using some Berry-Esseen type results.

34


Limit theorems for critical inhomogeneous branchingprocesses with immigration

Marton Ispany

A zero start inhomogeneous branching process with immigration (IBPI) (Xn)n∈Z+

is defined as

Xn =

Xn−1∑j=1

ξn,j + εn, n ∈ N, X0 = 0,

where ξn,j , εn : n, j ∈ N are independent non-negative integer-valued randomvariables such that ξn,j : j ∈ N are identically distributed for each n ∈ N.Assume that mn := Eξn,1, λn := Eεn, σ2

n := Varξn,1, b2n := Varεn are finite forall n ∈ N. The process (Xn)n∈Z+ is called (nearly) critical if mn → 1 as n→∞.Introduce the random step functions

X (n)(t) := Xbntc for t ∈ R+, n ∈ N.

We prove the following generalization of a result of Wei and Winnicki.

Theorem. Suppose that∑∞n=1 |mn − 1| < ∞; σ2

n → σ2 ≥ 0, λn → λ ≥ 0,

b2n → b2 ≥ 0, and n−2∑nk,j=1 E

(|ξk,j −mk|21|ξk,j−mk|>θn

)→ 0 for all θ > 0 as

n→∞. Then

n−1X (n) L−→ X as n→∞,

that is, weakly in the Skorokhod space D(R+,R), where(X (t)

)t∈R+

is the unique

solution of a stochastic differential equation (SDE)

dX (t) = λ dt+ σ√X+(t) dW (t), t ∈ R+,

with initial condition X (0) = 0, where x+ := maxx, 0 and (W (t))t∈R+ is astandard Wiener process.

References[1] Ispany, M., Pap, G. and van Zuijlen, M., Fluctuation limit of branching processeswith immigration and estimation of the means. Adv. Appl. Prob. 37 (2005), 523–538.[2] Gyorfi, L., Ispany, M., Pap, G. and Varga, K., Poisson limit of an inhomogeneousnearly critical INAR(1) model. Acta Sci. Math. Szeged 73(3-4) (2007), 789–815.[3] Wei, C.Z. and Winnicki, J., Estimation of the means in the branching processwith immigration. Ann. Statist. 18, (1990), 1757–1773.

35


Comparison of mixtures with non-mixtures1

Syeda Rabab Jafri

We are motivated by optimal bounds in Rosenthal and Khinthchine momentinequalities (so called extremal problems), such as Figiel, T., Hitczenko, P., Johnson,W.B., Schechtman, G. and Zinn, J. Extremal properties of Rademacher functionswith applications to the Khintchine and Rosenthal inequalities. Trans. Amer. Math.Soc. 349 no. 3, 997-1027. Mattner, Lutz. Mean absolute deviations of sample meansand minimally concentrated binomials. Ann. Probab. 31 (2003), no. 2, 914-925.

Possible extensions to non-commutative probability are also considered.

Asymptotic normality of kernel type regressionestimators for random fields

Zsolt Karacsony

The asymptotic normality of the Nadaraya-Watson regression estimator is studiedfor α-mixing random fields. The infill-increasing setting is considered, that is whenthe locations of observations become dense in an increasing sequence of domains.This setting fills the gap between continuous and discrete models. In the infill-increasing case the asymptotic normality of the Nadaraya-Watson estimator holds,but with an unusual asymptotic covariance structure. It turns out that this covari-ance structure is a combination of the covariance structures that we observe in thediscrete and in the continuous case.

References[1] Fazekas, I. and Chuprunov, A., Asymptotic normality of kernel type densityestimators for random fields. Stat. Inf. Stoch. Proc. 9 (2006), 161–178.[2] Karacsony, Zs. and Filzmoser, P., Asymptotic normality of kernel type regressionestimators for random fields. Manuscript.[3] Nadaraya, E. A., On estimating regression. Theor. Probability Appl., 9 (1964),141–142.[4] Watson, G. S., Smooth regression analysis. Sankhya. Ser. A 26 (1964), 359–372.

1Based on the joint work with Sergey Utev University of Nottingham

36


St. Petersburg portfolio games

Laszlo Gyorfi and Peter Kevei

We investigate the portfolio games, where the component games are iid ran-dom variables and have zero growth rate. For the St. Petersburg game we determinethe best constantly rebalanced portfolio. For several components we compute thegrowth rate numerically, and we determine the asymptotic growth rate as the num-ber of components tends to infinity.

Rate of Convergence for Certain Optimal StoppingProblems

Andrey O. Klesov

We consider an optimal stopping problem for a Levy process and apply it tofinding the fair price of an American type option.

Let (Xt, t ≥ 0) be a homogeneous Levy process, such that, a compoundPoisson process from the Levy–Khinchin decomposition

Xt =∑

k≤N(t)

ξk

is such that

1. ξk ≥ 0 almost surely,

2. ξk are independent identically distributed random variables,

3. Eξηk <∞ for some η > 0,

4. Nt is a simple Poisson process with intensity λ(t) and random variables ξjand Nt are independent.Denote

M = all stopping times τ ∈ [0,∞],

MT = all stopping times τ ∈ [0, T ].

Theorem. Let g(x) = (x+)η, η ∈ R+, q > 1, and

V (x) = supτ∈M

E(e−qtg(Xτ )Iτ<∞),

V (x, T ) = supτ∈MT

E(e−qtg(Xτ )

37


where IA is the indicator of an event A. Assume that E(X+1 ) < ∞. Then

∀x ∈ R ∀T > T0 ∃C(x), c:

0 ≤ V (x)− V (x, T ) ≤ C(x)ecT .

This result generalizes the results by Novikov and Shiryaev (2004) proved forthe discrete time.

Theorem (Sufficient conditions for a Poisson process). Let η > 0,q > 0. Let Π(t) be a Poisson process with intensity λ(t). If∫ ∞

1

e−ηqt maxλ(t), λη(t)dt <∞,

then for every T > 0

E

(supt≥T

Π(t)

eqt

)η<∞.

Experiences with Stochastic Modelling Tools

Attila Kuki

The lecture deals with the software tools for evaluating and analysing queue-ing networks. The main goal of the talk is to give some teaching experiences of thesetools.

The PEPSY-QNS (Performance Evaluation and Prediction SYstem for Queue-ing NetworkS) was developed in the 90’s at Erlangen University . It has a very sim-ple, user friendly interface and works under UNIX operating system. The PEPSY-QNS has an easy-to-use and fast non-graphical and a graphical (XPEPSY) versionas well. So far there are more than 50 analyzing methods built in the system.

At the beginning of the development of PEPSY-QNS the main question washow can the well-known methods of queueing network analysis be implementedand how can new methods be created and validated in the system. The latestapplications can use this reliable and validated tool for solving real problems ofdifferent areas.

A few years ago a new version of PEPSY (WinPepsy) was developed forWindows PCs. Though this tool can be used much more easier than PEPSY-QNS,and the outputs are more expressive, there exist several situations where the use ofPEPSY-QNS is recommended.

This paper demonstrates these situations, compares the work with the differ-ent PEPSY tools, investigates the differences in usability, and analyses the outputs.Some examples will be given for calculating the main characteristics (throughput,average service time, utilization, average response time, average waiting times etc.)by the help of the UNIX-based and Windows-based tools.

38


Performance Analyzes of a Proxy Cache Server Modelwith External Users

using the Probabilistic Model Checker PRISM 1

Tamas Berczes, Gabor Guta, Gabor Kusper,

Wolfgang Schreiner and Janos Sztrik

We report our experience with formulating and analyzing in the probabilisticmodel checker PRISM a web server performance model with proxy cache server andexternal users that was previously described in the literature in terms of classicalqueuing theory. We describe how to model a proxy cache server, a web server, in-ternal and external users in PRISM. The main contribution of the paper shows howto model in PRISM an input queue which might receive different type of messagesand has to place the answer in different output queues depending on the type of themessage.

References[1] T. Berczes, G. Guta, G. Kusper, W. Schreiner, and J. Sztrik. Analyzing WebServer Performance Models with the Probabilistic Model Checker PRISM. Technicalreport no. 08-17 in RISC Report Series, Johannes Kepler University Linz, Austria,2008.[2] T. Berczes and J. Sztrik. Performance Modeling of Proxy Cache Servers Journalof Universal Computer Science, 12(9):11391153, 2006.[3] PRISM Probabilistic Symbolic Model Checker. www.prismmodelchecker.org,2008.

On almost sure limiting behavior of weighted sums ofrandom fields

Zbigniew A. Lagodowski and Przemyslaw Matula

We study necessary and sufficient conditions for the almost sure convergenceof averages of independent random variables with multidimensional indices obtainedby certain summability methods.

1Supported by the Austrian-Hungarian Scientific/Technical Cooperation Contract HU13/2007.

39


Phonebook-centric social networks-dealingwith similarities

Peter Ekler and Tamas Lukovszki

The capabilities of mobile phones enable them to participate in popular so-cial network applications. In addition to the relations between the users of thesocial network, the phonebooks in the mobile phones also define social relations.Phonebook-centric social networks also provide a synchronization mechanism be-tween phonebooks of users and the social network. By the synchronization the goalis to identify the persons listed in the phonebook and the network, it means tofind similar entries and keep the data consistent. Based on the phonebook-centricsocial network implementation, called Phonebookmark, we investigate the structureof such networks. We experienced that the distribution of the number of similaritiesfollows a power law distribution, as well as the distribution of the in- and outdegreesof the users in the network.

We show how to estimate the total number of similarities in phonebook-centric social networks. We verify the estimation also with measurements.

Keywords: social networks, mobile phones, power law distributions.

Flexible Scheduling Discipline for Fixed-Size-PacketSwitched Networks

Tamas Marosits and Sandor Molnar

How to provide Quality of Service is one of the main question in the recentand future internet. The wide variety of services demand granularity while thehuge amount of traffic require robustness and simplicity in traffic control (e.g., callacceptance, routing, scheduling).

In our previous work ([1]) the Advanced Round Robin (ARR) scheduling al-gorithm for fix packet length packet switching networks was introduced. As its namesuggests ARR is a round robin-type scheduling method in which every connectionhas its own buffer and there is a known time limit between the service opportunitiesof these buffers. The main difference is that queues in ARR may be served morethan once in a cycle and the access periods of a flow are uniformly distributed dur-ing the service cycle. Considering the architecture of ARR scheduler we can easilyformulate worst case delay and the maximum difference between successive packetdepartures of the flows. Moreover, if we know the arrival process of the connectionseven average delay and average difference between successive packet departures canbe given.

40


The above mentioned results are based on the consideration, that we alreadyhave an optimal organized service cycle in which the service opportunities of a con-nection are uniformly distributed in the service cycle. Obviously, this can not bemade for every possible combination of traffic parameters and service requirementsif we want to have a work conserving scheduler. However, we can build subop-timal service cycles and can examine the difference between the optimal and thesuboptimal solutions.

In this paper we present some further important features of ARR. Thesefeatures make it possible to locate ARR in the broad list of known schedulers.

As several well known scheduling algorithms do belong to the class of LatencyRate servers (described in [3]) also does ARR which will be shown in this paper.According to the characterization of LR−servers new bounds can be given for theARR, however these bounds might be looser then those which were evaluated basedon the architecture of ARR. The ARR can be also characterized as a GuaranteedRate server and we calculate the error term to this characterization. We show thatARR is in the class of Latency Rate servers, therefore all results of LR−servers canbe applied for ARR.

We also show the connection between ARR and Generalized Processor Shar-ing (GPS) discipline. Presenting the relationship between ARR and GPS we will beable to take full advantages of results achieved in connection of GPS: for examplenew worst case guarantees can be formulated for single node case and a multi-nodeARR-scenario can be easily analyzed taking into account the results can be found inthe literature (e.g., [4], [2], [5]). Moreover, we also present the fairness-index of ARRwhich is one of the most important feature of schedulers. Calculating the fairnessof ARR it become comparable to the known schedulers and the applicability of themethod will be obvious.

These results can provide a good guideline framework for designers to applyARR and calculate its parameters and performance.

References[1] T. Marosits, S. Molnar, and J. Sztrik, ”CAC Algorithm Based on AdvancedRound Robin Method for QoS Networks”, in the proceedings of The 6th IEEE Sym-posium on Computers and Communications, pp. 266-274, Hammamet, Tunisia, 3-5July 2001.[2] A. K. Parekh and R. G. Gallager, ”A Generalized Processor Sharing Approach toFlow Control in Integrated Services Networks: The Multiple Node Case”,IEEE/ACMTransactions on Networking, vol. 2, no. 2, pp. 137-150, April 1994.[3] D. Stiliadis and A. Varma, ”Latency-Rate Servers: A General Model for Analysisof Traffic Scheduling Algorithms”, IEEE/ACM Transactions on Networking, Octo-ber 1998.[4] R. Szabo, P. Barta, J. Bıro and F. Nemeth, ”Non-Rate Proportional Weighting ofGeneralized Processor Sharing Schedulers”, in the Proceedings of GLOBECOM99,Rio de Janeiro, Brasil, December 1999.[5] P. Valente, ”Exact GPS simulation and optimal fair scheduling with logarithmiccomplexity”,IEEE/ACM Transactions on Networking, vol. 15, no. 6, pp. 1454-

41


1466, December 2007.

Invariance Principles and Functional AsymptoticConfidence Intervals for the Slope in Linear Structural

and Functional Error-in-Variables Models

Yuliya V. Martsynyuk

A modified least squares process (MLSP) is introduced in D[0, 1] for the slopein linear structural and functional error-in-variables models (EIVM’s). Sup-normapproximations in probability and, as a consequence, functional central limit theo-rems (FCLT’s) are established for a data-based self-normalized version of this MLSP.MLSP is believed to be a new type of object of study, and invariance principles for itconstitute new asymptotics, in EIVM’s. Moreover, the obtained data-based FCLT’sfor the MLSP open up new possibilities for constructing various asymptotic confi-dence intervals (CI’s) for the slope that we call functional asymptotic CI’s. Twospecial examples of such CI’s are given. The talk highlights some of the speakersselected publications.

Cramer-von Mises test for the Weibull and Paretodistributions 1

Gennadi V. Martynov

Let Xn = X1, X2, ..., Xn be the sample from the r.v. with the distributionfunction F (x), x ∈ R1. We will test the hypothesis

H0 : F (x) ∈ G = G(x, θ), θ = (θ1, θ2, ...θk)> ∈ Θ ⊂ Rk.

The set of the alternative distributions contains all another distributions. We willconsider the Cramer-von Mises statistic

ω2n = n

∫ ∞−∞

(Fn(x)−G(x, θn))2 dG(x, θn),

θn is an estimator of θ, Fn(x) is the empirical distribution function. The resultsbelow are applicable also to the Kolmogorov-Smirnov statistic.

1This research was partially supported by Russian foundation for fundamental research:09-01-00740-a.

42


The distribution of ω2 depends generally from unknown value θ0 of the pa-rameter θ and the distribution family G. Khmaladze [3] has proposed the method ofempirical process transformation for eliminate such dependance. We will use herethe traditional approach based on using of the statistic ω2

n. It was found in 1955(see [2]]) that the empirical process does not depend on unknown parameter for thefamily Q = Q((x−m)/σ), −∞ < x <∞, σ > 0.

We will propose here second class of the distribution family with analogousproperty: R = R((x/β)α), α > 0, β > 0, x ∈ X ⊂ [0,∞), where X is thesupport of the distribution R((x/β)α), R(z) is a distribution function with thesupport Z ⊂ [0,∞). Let r(z) = R′(z)). The Cramer-von Mises and Kolmogorov-Smirnov tests based on the empirical process ξn(x) =

√n(Fn(x) − R((x/β)α))),

where α and β are the ML estimates of α and β. The covariance function for thetransformed to (0, 1) limit Gauss process ξ(t) is:

K(t, τ) = min(t, τ)− tτ − (1/(B11B22 −B212))

×(B22s1(t)s1(τ)−B12(s1(t)s2(τ) + s2(t)s1(τ)) +B11s2(t)s2(τ)), t, τ ∈ (0, 1).

B11 =

∫Z

(z log z r′(z)

r(z)+ log z + 1

)2

r(z)dz, B22 =

∫Z

(z r′(z)

r(z)+ 1

)2

r(z)dz,

B12 =

∫Z

(z log z r′(z)

r(z)+ log z + 1

)(z r′(z)

r(z)+ 1

)r(z)dz,

s1(t) = r(R−1(t))R−1(t) log(R−1(t)), s2(t) = r(R−1(t))R−1(t). Hence, that thelimit distributions of the considered statistics do not depend from the parameters αand β.

The theory is applicable particularly for the the Pareto distribution

F (x) = 1− (x/β)−α, x ≥ β ≥ 0, α > 0, R(z) = 1− 1/z, Z = [β,∞],

(β has the supereffective estimstor) for two parametric Weibull distribution family

F (x) = 1− e−(x/β)−α , x ≥ 0, β ≥ 0, α > 0, R(z) = 1− e−z, Z = [0,∞],

for the power distribution function

F (x) = xα, x ∈ [0, 1], R(z) = z, Z = [0, 1].

The limit distribution of ω2n can be calculated exactly with using the meth-

ods described in [1, 4, 5]. These methods are applicable for tests with the empiricalprocess multiplying by the weight function ψ(t) = tv, v > −1.

References[1.] Deheuvels, P., Martynov, G. (2003) Karhunen-Loeve expansions for weightedWiener processes and Brownian bridges via Bessel functions. Progress in Probabil-ity., 55, 57–93. Birkhauser, Basel/Switzerland.[2.] Kac, M., Kiefer, J., Wolfowitz, J. (1955) On tests of normality and other testsof goodness-of-fit based on distance methods. Ann. Math. Statist., 30, 420–447.

43


[3.] Khmaladze, E.V. (1981) A martingale approach in the theory of parametricgoodness-of-fit tests. ” Theor. Prob. Appl.”, 26, 240–257.[4.] Martynov, G. V. The omega square tests. Moscow, ”Nauka” , 1979, 80pp.[5.] Martynov, G. V. (1994) Weighted Cramer-von-Mises test with estimated pa-rameters. LAD’2004: Longevty, Aging and Degradation Models, StPeterburg, 2,207–222

SLLN for random fields under restrictions on thebivariate dependence structure

Przemyslaw Matula

Let (Xn)n∈Nr be a family of identically distributed random variables indexedby r−dimensional lattice points n ∈ Nr. We study the necessary and sufficientconditions for the strong law of large numbers for such random fields under thefollowing condition imposed on the bivariate dependence structure:

CXi,Xj (u, v)− uv ≤ qijuv(1− u)(1− v),

where CXi,Xj (u, v) is the copula of Xi and Xj , i 6= j and qij ≥ 0 satisfy someadditional assumptions.

Optimal portfolios and stochastic dominancefor some bivariate distributions

Balazs Nyul

We study optimal portfolios where portfolios are optimised in expected util-ity. We fix certain utility functions and study the following question: under whatcondition will the proportion of an asset be larger (dominance) than the one of theother asset in the optimal portfolio. For this we consider and characterize particularbivariate distributions.

44


A strong approximation theorem for the estimation ofGARCH parameters

Laszlo Gerencser and Zsanett Orlovits

A fundamental tool in modeling financial time series, in particular stochasticvolatility processes is the so called GARCH model due to Engle and Bollerslev. Theestimation of the dynamics of a GARCH process is a basic and widely discussedproblem. Consistency results under the weakest possible conditions has been givenby Berkes et al. [2] using elementary methods.

The purpose of this paper is in a sense complementary: we present a strongapproximation result for GARCH processes with error terms, the moments of whichare bounded from above by a tight upper bound, under conditions that is requiredby the techniques that we use. In particular, we rely on the theory of stochasticapproximation with Markovian dynamics, detailed in [1]. The applicability of thistheory is far from trivial. See [3] for a related technical result. Using this strongapproximation result important properties of the estimator can be derived, that cannot be obtained by other techniques.

Our result is an extension of a similar result for ARMA processes, given in[4], with a uniform strong law of large numbers for the loglikelihood function beinga key element.

References[1] A. Benveniste, M. Mtivier and P. Priouret: Adaptive Algorithms and StochasticApproximations, Springer Verlag, Berlin, 1990.[2] I. Berkes, L. Horvath, P.S. Kokoszka: GARCH processes: structure and estima-tion,Bernoulli, 9 (2003) 201-217.[3] L. Gerencser and Zs. Orlovits: Lq-stability of products of blocktriangular sta-tionary random matrices, Acta Scientiarum Mathematicarum (Szeged) 74 (2008)927-944.[4] L. Gerencser: On the Martingale approximation of the estimation error of ARMAparameters, Systems and Control Letters 15 (1990) 417- 423.

45


Asymptotic behavior of maximum likelihood estimatorfor time inhomogeneous diffusions

Matyas Barczy and Gyula Pap

Parameter estimation for diffusion processes has been studied for a long time,see, e.g., the books of Kutoyants [3] and Bishwal [2].

First we consider a process (X(α)t )t∈[0,T ) given by a SDE

dX(α)t = αb(t)X

(α)t dt+ σ(t) dBt, t ∈ [0, T ),

with a parameter α ∈ R, where T ∈ (0,∞], b : [0, T )→ R, σ : [0, T )→ (0,∞) arecontinuous functions such that there exists t0 ∈ (0, T ) with the property b(t) 6= 0for all t ∈ [t0, T ), and (Bt)t∈[0,T ) is a standard Wiener process. We study

asymptotic behavior of the maximum likelihood estimator (MLE) α(X(α))t of α

based on the observation (X(α)s )s∈[0, t] as t ↑ T . We formulate sufficient conditions

under which √IX(α)(t)

(α

(X(α))t − α

) L−→ c

∫ 1

0Ws dWs∫ 1

0(Ws)2 ds

as t ↑ T ,

where IX(α)(t) denotes the Fisher information for α contained in the sample

(X(α)s )s∈[0, t], (Ws)s∈[0,1] is a standard Wiener process, and c = 1/

√2 or c =

−1/√

2. We also weaken the sufficient conditions due to Luschgy [4, Section 4.2]

under which√IX(α)(t)

(α

(X(α))t − α

)converges in distribution to the Cauchy

distribution. Furthermore, we give sufficient conditions so that the MLE of α isasymptotically normal with some appropriate random normalizing factor. Strong

consistency of the MLE α(X(α))t is also addressed.

Next we study a SDE

dY(α)t = αb(t)a(Y

(α)t ) dt+ σ(t) dBt t ∈ [0, T ),

with a perturbed drift satisfying a(x) = x + r(x), x ∈ R, where the function rfulfils the global Lipschitz condition and r(x) = O(1 + |x|γ) with some γ ∈ [0, 1).We give again sufficient conditions under which√

IY (α)(t)(α

(Y (α))t − α

) L−→ c

∫ 1

0Ws dWs∫ 1

0(Ws)2 ds

as t ↑ T .

We emphasize that our results are valid in both cases T ∈ (0,∞) andT =∞, and we develope a unified approach to handle these cases.

In a companion talk, Gyula Pap will study the asymptotic behavior of the

MLE α(X(α))t of α in special cases by calculating explicitly some Laplace trans-

forms.

46


Automatic Web Performance Simulation and Prediction,Based on a Combination of Time Series Analysis and

Queuing Network Simulation

Martin Pinzger

Performance is a key feature in many systems nowadays, especially in the fieldof web applications the performance decides about success or failure. A lot of sys-tems are developed with new techniques to ensure and test for adequate performancenowadays [2]. Nevertheless there are still a lot of applications where performanceaspects were not considered in the development process; here a system especiallyfor such applications will be drafted.

The substantial characteristics of the system which will be drafted now are;(a) the capability to automatically create a web performance simulation model (b)to automatically adjust the simulation model with the normal system load and(c) to automatically conduct trend analysis of the system under test (SUT). Theautomatic configuration, execution and prediction and the combination of simulationtechniques and time series / prediction techniques differ the drafted system fromprevious approaches like refer to in [1][3][8][10].

The system is built around the three key functionalities; these are monitoring,simulation and prediction. As the system should work completely automatic thestarting point and only necessary interaction point with the SUT is the monitoringcomponent. The monitoring component analyzes the SUT using software monitoring[9]. Three types of monitoring are distinct: active -, passive -, active and passive- software monitoring. With active - software monitoring it is possible to gain amaximum of information about the SUT at the cost of influencing the SUT e.g.Code Instrumentation. In opposition to this passive - software monitoring, gainsless information but has the advantage of no interference with the SUT e.g. packetsniffing. With active and passive - software monitoring both approaches shouldbe combined e.g. share a standard log over the network with the drafted system.Depending on the environment and the detail level required the drafted system iscapable of using one or all monitoring types in combination.

Based on the pieces of information about the SUT provided by the moni-toring component, a simulation model is generated automatically. Currently thedrafted system uses JSIM [4] and generates automatically a queuing network simu-lation model [9] supporting multi queue, multi server, and multi class models. Thegenerated simulation model is then automatically adjusted by the drafted system;for this the simulation model and the SUT are confronted with the same requests.These requests could be the normal - or artificial - system load, the adjustmentworks anyway. The generated responses (e.g. average response time) are then com-pared to each other and the simulation model is adjusted until the aimed accuracyis reached. For the determination of the parameters of the simulation model severaltechniques are used, the more telling are: Moving Average as a Reference, Medianas a first approach, ARMA (autoregressive moving average) as advanced prediction

47


method and ARMA with 60 seconds grouping where always 60 seconds of the basedata are grouped to one data point to increase the time frame which is consideredor predictions are made for.

Once the simulation model is adjusted the prediction component is startedin parallel to the continuous process of comparing and adjusting the simulationmodel to the SUT. The prediction component predicts the behavior of the SUTwith the help of the simulation model. To accomplish this, the prediction componentgenerates several scenarios and confronts the simulation model with these situations.Based on the outcome then predictions about the future behavior of the SUT aremade. This opens the opportunity that e.g. when the prediction suggests that inthe next hour the SUT has to prepare for a high load additional system resourcescould be reserved or long running batch jobs could be paused.

The drafted system is still work in progress only the main parts of the systemhave been realized; these are the monitoring, the simulation and the prediction part.In the area of long time prediction and pro active interaction with the SUT additionalwork should be done. Already published work dealing with this work could be foundin [5][6] and [7].

Preliminary results from the evaluation of the simulation model adjustmentstrategies resulting out of a system with artificial system load suggest a high po-tential in using ARMA as an adjustment method for simulation model adjustment.The analysis was done fully automated with the use of passive software monitoringand a multi class, single queue and single server simulation model.

Concerning the evaluation process of the drafted system the goal is to findhighly frequented web applications which are in productive usage, to test for ade-quate performance and accurate predictions. Additionally research concerning theinfluence of using different complexity levels for the simulation model should be an-alyzed e.g. a simulation model with one, two or three server.

References[1] Averill M. Law. Simulation Modeling and Analysis, Fourth Edition. McGraw-Hill, Inc. 2007[2] Daniel A. Menasce, Lawrence W. Dowdy, and Virgilio A. F. Almeida. Perfor-mance by Design: Computer Capacity Planning By Example. Prentice Hall PTR,Upper Saddle River, NJ, USA, 2004.[3] George K. Baah, Alexander Gray, and Mary Jean Harrold. On-line anomalydetection of deployed software: a statistical machine learning approach. In SO-QUA’f06: Proceedings of the 3rd international workshop on Software quality assur-ance, pages 70.77, New York, NY, USA, 2006. ACM.[4] Jennifer Hou. J-sim official, January 2005 http://sites.google.com/site/jsimofficial/[5] Martin Pinzger. Automated web performance analysis. In ASE, pages 513.516.IEEE, 2008.[6] Martin Pinzger: Automated web performance analysis, with a special focus onprediction. iiWAS 2008: 539-542[7] Martin Pinzger. Strategies for automated performance simulation model adjust-ment, preliminary results. In MONA+, 2008

48


[8] P. Schwarz and U. Donath. Simulation-based performance analysis of distributedsystems, 1997.[9] Raj Jain. The Art of Computer Systems Performance Analysis. John Wiley andSons, Inc., 1991.[10] S. Narayanan and S. McIlraith. Analysis and Simulation of Web Services. Com-puter Networks, 42(5):675.693, 2003.

Identification of almost unstable Hawkes processes

Vilmos Prokaj, Balazs Torma and Laszlo Gerencser

Self-exciting point processes, also called Hawkes processes are widely used tomodel credit events (defaults) on bond markets in financial mathematics. This is apoint process whose intensity is defined via feedback mechanism the input of whichis the past of the point process itself. The identification (calibration) of Hawkesprocesses is a hot research area. In this paper we consider Hawkes processes inwhich the feedback path is defined by a finite dimensional linear system with non-negative system matrices. This feedback system is well-defined if the integral of theimpulse response function of the feedback path is strictly less than 1.

We investigate the behavior of the Fisher information matrix of the (con-ditional) maximum-likelihood estimator when the system parameters approach theboundary of the stability domain. For standard Hawkes processes, with AR(1) im-pulse response, we prove that the inverse of the Fisher information-matrix tends tozero as we approach the stability boundary.

The extension of this result to the general case is still incomplete, but exten-sive numerical experiments indicate the validity of an appropriate extension, subjectto identifiability constraints. For standard Hawkes processes we calculate also thelimit distribution of the appropriately re-scaled intensity process. The latter resultcomplements nicely analogous results for almost unstable AR(1) processes by Gy.Pap and M.van Zuijlen.

49


On the statistical analysis of quantized GaussianAR (1) processes

Miklos Rasonyi

An often encountered problem is to determine signal parameters from roundedoff observations. The quantizer function q is specified as

q(x) = k for x ∈ [k − 1/2, k + 1/2), for all k ∈ Z.

We consider an AR(1) model on a probability space (Ω, F, P ):

Xn = α∗Xn−1 + εn, n ≥ 0 (1)

with a deterministic initial value X−1 and εn ∼ N(µ∗, (σ∗)2) i.i.d. Set Yn := q(Xn).

Let us fix the set of admissible parameters

Θ := (α, α)× (µ, µ)× (σ, σ),

with 0 < σ < σ, −1 < α < α < 1, µ < µ and assume that the true parameter

θ∗ = (α∗, µ∗, θ∗) lies in Θ. We define (Xθ, Y θ) analogously to (X,Y ) but with pa-rameter θ instead of θ∗.

The log-likelihood function is

Lθn(y0, ...yn) := lnP (Y θ0 = y0, ...Yθn = yn), y0, ..., yn ∈ Z.

Theorem 0.1. There is a deterministic function L : Θ → R such that for eachcompact Ξ ⊂ Θ, almost surely

1

nLθn(y0, ...yn) =→ L(θ), n→∞,

holds uniformly in θ ∈ Ξ.

This theorem forms the basis for showing the strong consistency of themaximum-likelihood estimates of θ∗.

50


On the random functional central limit theorems inL2[0, 1] with almost sure convergence

Zdzislaw Rychlik

In this paper we present functional random-sum central limit theorems inL2[0, 1] with almost sure convergence for independent nonidentically distributedrandom variables. We consider the case where the summation random indices andpartial sums are independent. In the past decade several authors have investigatedthe almost sure functional central limit theorems and related logarithmic limit the-orems for partial sums of independent random variables. We extend this theoryto almost sure versions of the functional random-sums central limit theorems inL2[0, 1]. We shall also present the almost sure random functional limit theoremsfor the empirical processes in L2[0, 1]. The presented results generalize or extend,to the random functional limit theorems in L2[0, 1] for sequences of independentnonidentically distributed random variables the main theorems presented by J. Turi(2002), M. Csorgo and S. Csorgo (1970, 1973), S. Csorgo (1974), G. A. Brosamler(1988) and P. Schatte (1988, 1991).

The main problem when proving this kind of results is the proof of the relativecompactness of the sequence of L2[0, 1] valued random elements. In order to solvethese difficulties we use the methods presented by P. E. Oliveira and Ch. Suquet(1988, 1995) and P. E. Oliveira (1990).

Closed form results for BMAP/G/1 vacationmodel with a class of service disciplines

Zsolt Saffer and Miklos Telek

The paper deals with the analysis of the BMAP/G/1 vacation model. We ap-ply a formerly introduced two-step methodology separating the analysis into servicediscipline independent and service discipline dependent parts. In this paper we in-vestigate the later analysis part for a class of service disciplines, for which a specificform functional equation can be established for the vector probability generatingfunction (vector GF) of the stationary number of customers at start of vacations.Such disciplines are e.g binomial-gated and the binomial-exhaustive ones.

We provide new results for the model with these disciplines. These are theclosed-form expressions of the vector GF of the stationary number of customers atstart of vacations and at arbitrary time.

Keywords: queueing theory, vacation model, BMAP, functional equation.

51


Outliers in INAR(1) models

Matyas Barczy, Marton Ispany, Gyula Pap,

Manuel Scotto, Maria Eduarda Silva

During the last decades there has been considerable interest in integer-valuedtime series models and a sizeable volume of work is now available in specializedmonographs. Motivation to include discrete data models comes from the need toaccount for the discrete nature of certain data sets, often counts of events, objectsor individuals. Examples of applications can be found in the analysis of time seriesof count data in many areas. Among the most successful integer-valued time seriesmodels proposed in the literature are the INteger-valued AutoRegressive model oforder p (INAR(p)). The statistical and probabilistic properties of the INAR(p)models have been studied by many authors.

Moreover, topics of major current interest in time series modeling are todetect outliers in sample data and to investigate the impact of outliers on the esti-mation of conventional ARIMA models. Motivation comes from the need to assessfor data quality and to the robustness of subsequent statistical analysis in the pres-ence of discordant observations. Fox (1972) introduced the notion of additive andinnovational outliers and proposed the use of maximum likelihood ratio test to de-tect them. His techniques have been generalized to other models for continuousvariables.

A related interesting problem, which has not yet been addressed, is to inves-tigate the impact of outliers on the parameter estimation for integer-valued autore-gressive models. This paper aims at giving a contribution towards this direction.In particular, we consider the problem of estimating the parameters of INAR(1)models contaminated by additive and innovational outliers starting from a generalinitial distribution (having finite second or third moments). We suppose that thetime points of the outliers are known, but their sizes are unknown.

Under the assumption that the second moment of the innovation distributionis finite, we prove assymptotic properties for the Conditional Least Squares (CLS)estimators for the means of the offspring and innovation distributions.

52


Performance Modeling Tools1

Janos Sztrik

This paper deals with the role of performance modeling tools. It introduces4 major tool developer centers and shows how a given tool can be applied to reli-ability investigations of finite-source retrial queueing system in steady state. Somenumerical examples are given demonstrating the effect of failure and repair rates ofserver on the mean response time of sources.

Keywords: performance modeling tools, finite-source, retrial queueing systems,telecom munication system

References[1.] Artalejo, J. and C. A. Gomez (2008). Retrial Queueing Systems. Berlin:Springer.[2.] Begain, K., G. Bolch, and H. Herold (2001). Practical performance modeling,application of the MOSEL language. Boston: Kluwer Academic Publisher.[3.] Mieghem, P. (2006). Performance Analysis of Communication Networks andSystems. Cam bridge: Cambridge University Press.

Long-range dependence in thirdorder for non-Gaussian time series

Gyorgy Terdik

If a time series is not Gaussian all its higher order cumulants are necessary forthe description of the dependence structure. The independence, for instance, impliesand implied by that each higher order cumulant is zero, except all variables in thecumulant are the same. Although non-Gaussianity and long-range dependence havebeen observed in many areas, in network traffic, in asset returns and exchange ratedata etc., there are no results neither on statistics of higher order nor on higherorder spectra of non-Gaussian and long-range dependent time series.

The object of this paper is to define the long-range dependence (LRD) inthird order of a time series and to investigate the third order properties of some wellknown long-range dependent non-Gaussian series. We define the third order LRDin terms of the third order cumulants and of the bispectrum. Both the third order

1Research is partially supported by Hungarian Scientic Research Fund-OTKAK60698/2005

53


cumulants and the bispectrum are symmetric, therefore we consider their valueson the principal domains only. The definition of the third order LRD is given inpolar coordinates, since the origin and the x−axis have some particular importance.Besides the primary singularity of the bispectrum at the origin, the singularityon the whole x−axis is allowed. Similarly, not only the radial decay of the thirdorder cumulants at infinity is considered but its behavior when it is approachingthe x−axis on the ’circle with infinite radius’ as well. The marginal bispectrum andthe third order cumulants on the x−axis couple these properties of the bispectrumand the third order cumulants. We consider three basic non-Gaussian models; theFractionally integrated noise, H2−process and ∆LISDLG process. These modelsserve as prototypes for the third order LRD.

Finally we summarize the results and put some conjectures for further sub-jects of investigations.

A general method to obtain the rate of convergence inthe strong law of large numbers and its applications

Tibor Tomacs

It is well-known that the Hajek-Renyi inequality (see [3]) is a generalizationof the Kolmogorov inequality. In [5] we show that Kolmogorovs inequality impliesa certain Hajek-Renyi type inequality. Using this fact we give a general method toobtain strong laws of large numbers. Actually our method is the same as the oneapplied in Fazekas and Klesov [1] and Fazekas et al. [2] but here we use probabilitiesinstead of moments. In the proof we follow the lines of [1].

Our theorem offers a general tool: if a maximal inequality is known for acertain sequence of random variables then one can easily obtain a strong law of largenumbers. Our scheme helps to find the conditions and the normalizing constants.

We apply our theorem to give alternative proofs for some known strong lawsof large numbers. We deal with associated, negatively associated random variablesand demimartingales.

Sung, Hu and Volodin [8] introduced a new method for obtaining convergencerate in the strong law of large numbers, by using the approach of Fazekas and Klesov[1]. This result generalizes and sharpens the method of Hu and Hu [4].

In [6] we give a general method by using a Hajek-Renyi type inequality forthe probabilities, which sharpens the result of Sung, Hu and Volodin [8].

In [7] we apply this method for mixingales and superadditive structures.

References[1] Fazekas, I. and Klesov, O., A general approach to the strong laws of large num-bers, Theory of Probab. Appl., 45/3 (2000) 568583.[2] Fazekas, I., Klesov, O. I., Noszaly, Cs., Tomacs, T., Strong laws of large numbers

54


for sequences and fields, (Proceedings of the Third Ukrainian- Scandinavian Con-ference in Probability Theory and Mathematical Statistics 812 June 1999. Kyiv,Ukraine) Theory of Stochastic Processes, Vol.5 (21) no. 34 (1999) 91104.[3] Hajek, J. and Renyi, A., Generalization of an inequality of Kolmogorov, ActaMath. Acad. Sci. Hungar. 6 no. 34 (1955) 281283.[4] Hu, S., Hu, M., A general approach rate to the strong law of large numbers, Stat.and Prob. Letters, 76 (2006) 843851.[5] Tomacs, T. and Lıbor, Zs., A Hajek-Renyi type inequality and its applications,Annales Mathematicae et Informaticae, 33 (2006) 141149.[6] Tomacs, T., A general method to obtain the rate of convergence in the strong lawof large numbers, Annales Mathematicae et Informaticae, 34 (2007) 97102.[7] Tomacs, T., Convergence rate in the strong law of large numbers for mixin-gales and superadditive structures, Annales Mathematicae et Informaticae, 35 (2008)147154.[8] Sung, S.H., Hu, T.-C., Volodin, A., A note on the growth rate in the Fazekas-Klesov general law of large numbers and on the weak law of large numbers for tailseries, Publicationes Mathematicae Debrecen, 73/1-2 (2008) 110.

Hidden Markov Model Based SpeakerDependent and Adaptive Training of Hungarian

Text-to-Speech System

Balint Toth and Geza Nemeth

Hidden Markov Models (HMM) can be applied in numerous fields: machinetransition, gene prediction, partial discharges, speech recognition, cryptoanalisis,machine translation, etc. In the last decade hidden Markov model based speechsynthesis although got in focus as a new aspect of generating human soundingartificial speech.

The model learns the spectral, excitation and timing parameters of a ratherlarge speech corpora from one (speaker dependent training) or from various (adap-tive training) speakers. In the second case a very small speech corpus is enough toadapt the voice characteristics to this target speaker. The parameters are stored inHMM databases and with a simple voice decoder method they are transformed intohigh quality human sounding artificial voice.

The current study describes the basics of speaker dependent and adaptiveHMM based speech synthesis and introduces the steps of creating a Hungarian HMMbased Text-to-Speech system and shows the results of a simple mean opinion scorelistening test.

55


Limit theorems for the longest run

Jozsef Turi

Consider N tossings of a coin. Let p be the probability of head and q = 1−pthe probability of tail. We study the longest run that is the longest run of pure headon pure tails. Let µ∗(N) denote the length of the longest run in the first N trials.We have the following almost sure limit theorem.

limn→∞

1

logn

n∑i=1

1

iI(µ∗(i)− Log i < t) =∫ t+1

texp

[−(

12

)y]dy if p = 1

2.∫ t+1

texp [−qpy] dy if p > 1

2.

Here Log denotes the logarithm of base 1/p.In Mori (1993) an almost sure limit theorem was obtained for the longest

head run. Our result is a version of Mori’s theorem.

References[1] Mori, T.F. (1993). The a.s. limit distribution of the longest head run. Can. J.Math., 45(6), 1245–1262.

Accelerated Failure Time Modelunder measurement error

Elena Usoltseva

Consider the Accelerated Failure Time Model with i.i.d. and zero meanerror. Some lifetimes may be censored, in that only a lower bound for the lifetimeis recorded. The distribution of lifetimes depends on unknown parameter, whichis estimated. The distribution of censor does not depend of this parameter. Ameasurement error is present in our model.

Under the presence of measurement errors, the adjusted unbiased CorrectedScore estimating function for messy lifetime was constructed early in Thomas Au-gustin’s works.

In this report we prove that this estimating function yields a consistent esti-mator under bounded censor. We consider a sample of n nonnegative i.i.d. randomvariables of lifetimes. Under n follows to infinity asymptotic property of estimatoris considered.

First there is proved that constructed estimator in the case of known distri-bution of censor is consistent. But usually censoring distribution is unknown. It

56


can be estimated from data by Kaplan-Meier estimator. When we use Kaplan-Meierestimator, the failure and censoring will be alternated.

Then we prove the consistency of estimator in the case of unknown distribu-tion of censor under measurement error. Suppose that the distribution was estimatedfrom data by Kaplan-Meier estimator. Therefore, the estimator that defined as theroot of the constructed estimation equation remains consistent after estimating ofcensoring distribution by Kaplan-Meier method.

On consistency of R-estimatorsin errors-in-variables models

Silvelyn Zwanzig

Assume a functional errors-in-variables model

yi = g(ξi, β0) + εi, xi = ξi + δi, i = 1, .., n

with mutually i.i.d. errors. Rank estimators are defined as a minimum of a sum ofresiduals Zi weighted by rank scores an(Ri), where Ri is the rank of Zi

β ∈ arg min

n∑i=1

Zian(Ri).

In the talk we study R estimators related two different distances between the obser-vation points (xi, yi) and the curve (ξ, g(ξ, β)): the vertical distance(yi − g(ξi, β))2 and the orthogonal distance

d2(xi, yi) = minξ

[(yi − g(ξ, β))2 + (xi − ξ)2

].

In the simple linear errors in variables model g(ξ, β) = ξβ the naive (vertical)

R-estimator βnaive is biased while the orthogonal R-estimator βorth is consistent.In the general nonlinear model the behavior of βnaive is studied for small

variance asymptotics.

References[1] J. Hajek (1968) Asymptotic Normality of Simple Linear Rank Statistics UnderAlternatives. Ann. Math. Stat. 39, 325-346[2] Jaeckel, L.A. (1972) Estimating regression coefficients by minimizing the disper-sion of the residuals. Ann.Math.Stat.,43, 1449-1458[3] Kuljus, K. and Zwanzig, S (2008). Asymptotic linearity of linear rank statisticsin the case of symmetric heteroscedastic variables, U.U.D.M Report 2008:32

57

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1. Faculty of Informatics – Registration

2. Chemical Building – Presentations

3. Life Science Building – Presentations

4. Main building of Debrecen University

5. Canteen

6. Tram Station

7. I. Kossuth Lajos Student Hostel

8. II. Kossuth Lajos Student Hostel

9. III. New Kossuth Lajos Student Hostel

10. Library

11. Botanical Garden

12. Medical and Health Science Center

2

1 3

4

5

6

7

8

9

10

11

12

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International Conference Probability and Statistics with ... · Probability and Statistics with Applications ... computer science and informatics. ... International Conference Probability