Pliska Studia Mathematica Bulgarica
Volume 13, 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_{n}V_{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.