Pliska Studia Mathematica Bulgarica

Volume 13, 2000

Proceedings of the 9th International Summer School on Probability Theory and Mathematical Statistics Sozopol, 1997

GUEST EDITOR: N. M. Yanev

Sofia, 2000

C O N T E N T S

• Neuts, M. Algorithmic Methods in Queues and in the Exploration of Point Processes. (pp. 5-13)
• Rouault, A. Large Deviations and Branching Processes. (pp. 15-38)
• Waymire, E. Scaling and Multiscaling Exponents in Networks and Flows. (pp. 39-56)
• Dimov, I. Monte Carlo Algorithms for Linear Problems. (pp. 57-77)
• Jacob, C. Asymptotic behaviour of a supercritical Galton-Watson process with controlled binomial migration. (pp. 79-90)
• Rukhin, A. Making Multiple Decisions Adaptively. (pp. 91-115)
• Dimov, I., Gurov, T. Monte Carlo Algorithm for Solving Integral Equations with Polynomial Non-linearity. Parallel Implementation. (pp. 117-132)
• Dimov, I., Karaivanova, A. Statistical Numerical Methods for Eigenvalue. Parallel Implementation. (pp. 133-149)
• Kolev, N., Minkova, L. A Characterization of the Negative Binomial Distribution. (pp. 151-154)
• Marintcheva, M. On Some Sufficient Conditions for High Breakdown Point of ML estimators. (pp. 155-160)
• Mitov, K. The maximal number of particles in a branching process with state-dependent immigration. (pp. 161-167)
• Mutafchiev, L. Large Distinct Part Sizes in a Random Integer Partition. (pp. 169-172)
• Nitcheva, D., Yanev, N. A System for Simulation and Estimation of Branching Processes. (pp. 173-178)
• Pancheva, E. On Self-Similar Extremal Processes. (pp. 179-193)
• Stoimenova, E. Evaluating the Risk in Selection Problems. (pp. 195-198)
• Yanev, G., Yanev, N. Limit Theorems for Branching Processes with Random Migration Components. (pp. 199-205)

A B S T R A C T S

ALGORITHMIC METHODS IN QUEUES AND IN THE EXPLORATION OF POINT PROCESSES
Marcel F. Neuts marcel@tucson.sie.arizona.edu

1991 Mathematics Subject Classification:60G55, 60K25, 60K20, 60J10.
Key words: Markovian arrival processes, queues, matrix-analytic methods, algorithmic probability.

This is a review of methodology for the algorithmic study of some useful models in point process and queueing theory, as discussed in three lectures at the Summer Institute at Sozopol, Bulgaria. We provide references to sources where the extensive details of this work are found. For future investigation, some open problems and new methodological approaches are proposed.

LARGE DEVIATIONS AND BRANCHING PROCESSES
Alain Rouault rouault@math.uvsq.fr

1991 Mathematics Subject Classification: 65F10, 60J80.
Key words: large deviations, branching processes, branching random walk, branching brownian motion.

These lecture notes are devoted to present several uses of Large Deviation asymptotics in Branching Processes. Two types of applications of large deviations in supercritical Galton-Watson processes are presented. The first one is a large deviation theorem for the Lotka-Nagaev estimator of the mean. The second one is particularly nice. It uses the "self-similarity" of the r.v. W limit of the process to give tail behaviours of the distribution of W (in 0 and in \infty).

SCALING AND MULTISCALING EXPONENTS IN NETWORKS AND FLOWS
Edward C. Waymire

1991 Mathematics Subject Classification: 60G48, 60J80.
Key words: multiplicative cascades, tree networks, networks flow extrems.

The main focus of this paper is on mathematical theory and methods which have a direct bearing on problems involving multiscale phenomena. Modern technology is refining measurement and data collection to spatio-temporal scales on which observed geophysical phenomena are displayed as intrinsically highly variable and intermittant heirarchical structures,e.g. rainfall, turbulence, etc. The heirarchical structure is reflected in the occurence of a natural separation of scales which collectively manifest at some basic unit scale.

MONTE CARLO ALGORITHMS FOR LINEAR PROBLEMS
Ivan Dimov, dimov@copern.acad.bg

1991 Mathematics Subject Classification: 65C05, 65U05.
Key words: Monte Carlo algorithms, linear problems, boundary value problem, efficiency estimator Markov chain, parallel algorithms.

In this paper the Monte Carlo numerical algorithms are considered which are usually used for solving deterministic problem by modeling random variables or random fields. The main idea here is to construct some artificial random process and to prove that the mathematical expectation of the process is equal to the unknown solution of the problem or to some functional of the solution. Usually, there are more than one possibility to create such an artificial process. After finding the process one needs to define an algorithm for computing realizations of the random variable. Usually, the random variable can be considered as a weight of a random process (usually, a Markov process). Then, The Monte Carlo algorithm for solving the problem under consideration consists in simulation the Markov process and computing the realizations of the random variables.

ASYMPTOTIC BEHAVIOUR OF A SUPERCRITICAL GALTON-WATSON PROCESS WITH CONTROLLED BINOMIAL MIGRATION
Christine Jacob christine.Jacob@jouy.inra.fr

1991 Mathematics Subject Classification:60J80, 62F12, 62P10.
Key words: Galton-Watson, supercritical, migration, binomial, size-dependent.

This paper considers a native population in which each individual can mutate with the same probability or the general epidemiologic problem where each individual of the population can catch a disease with the same probability. A population is only partially observed at each generation: for example, the population is in a volume V_n at generation n and the observation is done by means of an aliquot v_n, this aliquot being removed after observation. In this case each individual can be observed with the probability p_n = v_nV_n^-1.

The population of individuals who change (by mutation or disease or observation can be considered as an emigrating population. Systematic emigration can easily lead to the extinction of the population except when the emigration is controlled and the native process is supercritical.

MAKING MULTIPLE DECISIONS ADAPTIVELY
Andrew L. Rukhin rukhin@math.umbc.edu

1991 Mathematics Subject Classification: 60C05, 62C20, 62C25.
Key words: multiple decisions, adaptation, exponential families, consistancy.

The asymptotic behavior of multiple decision procedures is studied when the underlying distributions depend on an unknown nuisance parameter. An adaptive procedure must be asymptotically optimal for each value of this nuisance parameter, and it should not depend on its value. A necessary and sufficient condition for the existence of such a procedure is derived. Several examples are investigated in detail, and possible lack of adaptation of the traditional overall maximum likelihood rule is discussed.

MONTE CARLO ALGORITHM FOR SOLVING INTEGRAL EQUATIONS WITH POLYNOMIAL NON-LINEARITY. PARALLEL IMPLEMENTATION
Ivan T. Dimov dimov@amigo.acad.bg
Todor V. Gurov gurov@iscbg.acad.bg

1991 Mathematics Subject Classification: 65C05, 65U05.
Key words: Monte Carlo algorithms, Markov chain, parallel algorithm.

An iterative Monte Carlo algorithm for evaluating linear functionals of the solution of integral equations with polynomial non-linearity is proposed and studied. The method uses a simulation of branching stochastic processes. It is proved that the mathematical expectation of the introduced random variable is equal to a linear functional of the solution. The algorithm uses the so-called almost optimal density function. Numerical examples are considered. Parallel implementation of the algorithm is also realized using the package ATHAPASCAN as an environment for parallel realization. The computational results demonstrate high parallel efficiency of the presented algorithm and give a good solution when almost optimal density function is used as a transition density.

STATISTICAL NUMERICAL METHODS FOR EIGENVALUE. PARALLEL IMPLEMENTATION
Ivan T. Dimov dimov@amigo.acad.bg
Aneta Karaivanova anet@amigo.acad.bg

1991 Mathematics Subject Classification: 65C05, 65U05.
Key words: Monte Carlo algorithms, eigenvalue, efficiency estimator, Markov chain, parallel algorithm.

The problem of evaluating the smallest eigenvalue of real symmetric matrices using statistical numerical methods is considered. Two new almost optimal Monte Carlo algorithms are presented: Resolvent Monte Carlo algorithm (RMC). The algorithm uses Monte Carlo iterations by the resolvent matrix and includes parameter controlling the rate of convergence; Monte Carlo algorithm with inverse iterations (MCII). Numerical tests are performed for a number of large sparse symmetric test matrices. The tests are performed on supercomputers Intel-PARAGON (which is a distributed memory multicomputer) and CRAY Y-MP C92A (a two-processor vector machine). Some information about the parallel efficiency of the algorithms is obtained, showing that the algorithms under consideration are well-parallized and well-vectorizable.

A CHARACTERIZATION OF THE NEGATIVE BINOMIAL DISTRIBUTION
Nikolay Kolev nkolev@math.bas.bg
Leda Minkova

1991 Mathematics Subject Classification: 60E02.
Key words: negative binomial, characterization.

In this article a characterization of the negative binomial distribution related to random sums is obtained which is motivated by the geometric distribution characterization given by Khalil et al. (1991). An interpretation in terms of an unreliable system is given.

ON SOME SUFFICIENT CONDITIONS FOR HIGH BREAKDOWN POINT OF ML ESTIMATORS
Maya Marintcheva, telecom@math.bas.bg

1991 Mathematics Subject Classification: 62F10, 62F35.
Key words: robust statistics, high breakdown point.

High breakdown point estimators LME(k) and LTE(k) for location and scale are obtained for symmetrical exponentially decreasing density family. A high breakdown point for LME and LTE is obtained for j(z) = O(e^{- azb}); a is a positive constant and b lies between 0 and 1. A contra example in case of j(z) = 1/z^p demonstrates the need of exponential decrease.

THE MAXIMAL NUMBER OF PARTICLES IN A BRANCHING PROCESS WITH STATE-DEPENDENT IMMIGRATION
Kosto V. Mitov kmitov@af-acad.bg

1991 Mathematics Subject Classification: 60J80, 60K05.
Key words: branching processes, state-dependent immigration, maximal number of particles, regenerative processes.

The limiting behavior of the maximal number of particles in the first n generations of a Bienayme-Galton-Watson branching process with immigration in the state zero is studied.

LARGE DISTINCT PART SIZES IN A RANDOM INTEGER PARTITION
Ljuben R. Mutafchiev mutafch@math.bas.bg

1991 Mathematics Subject Classification: 05A17, 60C05, 60F05.
Key words: random integer partitions, limit theorems.

A limit theorem for the number of large part sizes in a random and uniform integer partition is proved. The weak limit turns out to be a Gaussian one.

A SYSTEM FOR SIMULATION AND ESTIMATION OF BRANCHING PROCESSES
Daniela Nitcheva daniela@fmi.uni-sofia.bg
Nickolay M. Yanev yanev@math.bas.bg

1991 Mathematics Subject Classification: 60J80, 60J85, 62M05.
Key words: branching processes, non-parametric estimation, simulation.

A computer code system for simulation and estimation of branching processes is proposed. Using the system, samples for some models with or without migration are generated. Over these samples we compare some properties of various estimators.

ON SELF-SIMILAR EXTREMAL PROCESSES
Elisaveta I. Pancheva pancheva@math.bas.bg

1991 Mathematics Subject Classification: 60G18, 60G52, 60G70.
Key words: multivariate extremal processes, self-similarity, homogeneous max-increments, weak convergence.

Given an extremal process and a proper sequence of non-linear time-space changes, we study the limit behaviour of the sequence of the time-space changed process under a regularity condition on the norming sequence and asymptotic negligibility of the max-increments. The limit class consists of selfsimilar extremal processes. Their univariate marginals are max-selfdecomposable. If additionaly the initial extremal process has has homogeneous max-increments, then the limit process is max-stable.

EVALUATING THE RISK IN SELECTION PROBLEMS
Eugenia Stoimenova jeni@math.bas.bg

1991 Mathematics Subject Classification: 62F07.
Key words: partial rankings, indifference zone formulation, least favourable configuration, loss functions, invariance, risk function.

A fixed sample size procedure for selecting the t best populations is considered. The probability requirement is set to be satisfied under the indifference zone formulation. In order to minimize the average losses from misclassification, we use loss function which is sensitive to the number of misclassifications. The upper bound of the corresponding risk is derived for location parameter distributions. The risk function for the Least Favorable Configuration is derived in an integral form for a large class of distribution functions.

LIMIT THEOREMS FOR BRANCHING PROCESSES WITH RANDOM MIGRATION COMPONENTS
George P. Yanev gyanev@tarski.math.usf.edu
Nickolay M. Yanev yanev@math.bas.bg

1991 Mathematics Subject Classification: 60J80.
Key words: branching processes, random migration, emigration and immigration, limit theorems.

Due to the migration component the results presented in this paper reveal new effects in the asymptotic behavior of branching processes in comparison with the classical Kolmogorov and Yaglom's results. One can distinguish three different types of asymptotics for the critical process with migration depending on the relation between of emigration and immigration.