


Vol 214, No 4 (2016)
- Year: 2016
- Articles: 15
- URL: https://journal-vniispk.ru/1072-3374/issue/view/14732
Article
Mikhail Iosifovich Gordin. on the Occasion of the 70 Anniversary Birthday



Probabilistic Model for the Lotka-Volterra System with Cross-Diffusion
Abstract
Two approaches that allow to construct a probabilistic representation of a generalized solution of the Cauchy problem for a system of quasilinear parabolic equations are proposed. The system under consideration describes a population dynamics model for a prey-predator population. The stochastic problem associated with this parabolic system is presented in two forms, which give a way to derive the required probabilistic representation. Bibliography: 16 titles.



Distribution of Functionals of Special Diffusions with Jumps
Abstract
The paper deals with special class of diffusions with jumps. For the traditional class of such diffusions, the jumps occur at the moments corresponding to the moments of jumps of a Poisson process. The position at the moment of a jump can be arbitrary. A description of the traditional class of diffusions with jumps is well known. A natural generalization of this class and many other results are also given here. In the present paper, we consider diffusions, for which the position of diffusion in any moment of jump takes a finitely many values. Such moments, for example, are the first exit time from an interval, the moment inverse to the diffusion local time or the minimum of inverse local times. The results of interest are those that allow one to compute the distributions of various functionals of diffusion with jumps. For a diffusion, in particular for the Brownian motion, the results of M. Kac are of key importance for development of the theory of the distributions of integral functionals.



On Limit Theorem in Some Service Systems
Abstract
The aim of the paper is to study a service system model introduced by I. Kaj and M. Taqqu. A limit theorem for the process of integral workload on the service system is proved. This theorem generalizes the corresponding result of I. Kaj and M. Taqqu, because the weak convergence in the Skorokhod space is established.



On the Littlewood–Offord Problem
Abstract
The paper deals with studying a connection between the Littlewood–Offord problem and estimating the concentration functions of some symmetric infinitely divisible distributions. Some multivariate generalizations of Arak’s results (1980) are given. They establish a relationship of the concentration function of the sum and arithmetic structure of supports of the distributions of independent random vectors for arbitrary distributions of summands. Bibliography: 21 titles.



Asymptotically Efficient Importance Sampling for Bootstrap
Abstract
The Large Deviation Principle is proved for the conditional probabilities of moderate deviations of weighted empirical bootstrap measures with respect to a fixed empirical measure. Using this LDP for the problem of calculation of moderate deviation probabilities of differentiable statistical functionals, it is shown that the importance sampling based on influence function is asymptotically efficient.



On Estimation of the Intensity Density Function of a Poisson Random Field Outside the Observation Region
Abstract
A Poisson random field with intensity density function \( \frac{\leftthreetimes (x)}{\varepsilon } \) is observed in a bounded region G ⊆ ℝd. It is supposed that the unknown function ⋋ belongs to a known class of entire functions. The parameter ε is supposed to be known. The problem is to estimate the value ⋋(x) at the points x /∉ G. An asymptotic setup of the problem as ε → 0 is considered. Bibliography: 13 titles.



On Stochastic Models of Service System with Dependent Process Characteristics
Abstract
A generalization of a service system model introduced by I. Kaj and M. Taqqu is considered. Unlike the original model, the unnatural assumption on independence between the duration and required resources quantity of a service process is dropped. A number of limit theorems for the process of integral workload is presented. Among the considered limit processes are the Winer process, fractional Brownian motion, and stable Lévy process.






Nonprobabilistic Infinitely Divisible Distributions: The Lévy-Khinchin Representation, Limit Theorems
Abstract
Properties of generalized infinitely divisible distributions with Lévy measure \( \varLambda (dx)=\frac{g(x)}{x^{1+\upalpha}}dx, \) α ∈ (2, 4) ∪ (4, 6) are studied. Such a measure is a signed one and, hence, is not a probability measure. It is proved that in some sense these signed measures are the limit measures for the distributions of the sums of independent random variables. Bibliography: 6 titles



Probabilities of Small Deviations of the Weighted Sum of Independent Random Variables with Common Distribution That Decreases at Zero Not Faster Than a Power
Abstract
The paper presents estimates of small deviation probabilities of the sum \( {\displaystyle \sum_{j\ge 1}{\leftthreetimes}_j{X}_j} \) , where {⋋j} are positive numbers and {Xj} are i.i.d. positive random variables satisfying weak restrictions at zero and infinity. Bibliography: 16 titles.



The Mackenhoupt Condition and an Estimating Problem
Abstract
The paper considers a connection between weighted norm inequalities for the Hilbert transform with matrix valued weights and an estimating problem. A connection of the vector Muckenhoupt condition on the spectral density of the stationary noise and a possibility to transform a difficult estimating problem to another well-studied problem is established. Bibliography: 12 titles



Lattice Point Problem and Questions of Estimation and Detection of Smooth Multivariate Functions
Abstract
Let Nd(m) be the number of points of the integer lattice that belong to a d-dimensional ball of radius m (in the l1- and l2-norms). The aim of the paper is to study the asymptotic behavior of Nd(m) as d → ∞, m → ∞. It is shown that if d tends to infinity much faster than m, then the asymptotic is different from the asymptotic volume of a d-dimensional ball of radius m. Bibliography: 6 titles.



Final Distribution of a Diffusion Process with Final Stop
Abstract
A one-dimensional diffusion process is considered. The characteristic operator of this process is assumed to be a linear differential operator of the second order with negative coefficient at the term with zero derivative. Such an operator determines the measure of a Markov diffusion process with break (the first interpretation), and also the measure of a semi-Markov diffusion process with final stop (the second interpretation). Under the second interpretation, the existence of the limit of the process at infinity (the final point) is characterized. This limit exists on any interval almost surely with respect to the conditional measure generated by the condition that the process never leaves this interval. The distribution of the final point expressed in terms of two fundamental solutions of the corresponding ordinary differential equation, and also the distribution of the instant final stop are derived. A homogeneous process is considered as an example.



On an Approximation for the Solutions of Some Evolution Equations by the Expectations of Random Walks Functionals
Abstract
The paper deals with some problems concerning probabilistic representation and probabilistic approximation for solution of the Cauchy problem for the family of equations \( \frac{\partial u}{\partial t}=\frac{\sigma^2}{2}\varDelta u \) with complex parameter σ such that Reσ2 ≥ 0. This family coincides with the heat equation if Imσ = 0, and with the Schrӧdinger equation if Reσ2 = 0. Bibliography: 5 titles


