Open Access Open Access  Restricted Access Access granted  Restricted Access Subscription Access

Vol 214, No 4 (2016)

Article

Mikhail Iosifovich Gordin. on the Occasion of the 70 Anniversary Birthday

Journal of Mathematical Sciences. 2016;214(4):423-424
pages 423-424 views

Probabilistic Model for the Lotka-Volterra System with Cross-Diffusion

Belopolskaya Y.I.

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.

Journal of Mathematical Sciences. 2016;214(4):425-442
pages 425-442 views

Distribution of Functionals of Special Diffusions with Jumps

Borodin A.N.

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.

Journal of Mathematical Sciences. 2016;214(4):443-455
pages 443-455 views

On Limit Theorem in Some Service Systems

Garai E.S.

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.

Journal of Mathematical Sciences. 2016;214(4):456-466
pages 456-466 views

On the Littlewood–Offord Problem

Eliseeva Y.S., Zaitsev A.Y.

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.

Journal of Mathematical Sciences. 2016;214(4):467-473
pages 467-473 views

Asymptotically Efficient Importance Sampling for Bootstrap

Ermakov M.S.

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.

Journal of Mathematical Sciences. 2016;214(4):474-483
pages 474-483 views

On Estimation of the Intensity Density Function of a Poisson Random Field Outside the Observation Region

Ibragimov I.A.

Abstract

A Poisson random field with intensity density function \( \frac{\leftthreetimes (x)}{\varepsilon } \) is observed in a bounded region G ⊆ d. 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.

Journal of Mathematical Sciences. 2016;214(4):484-492
pages 484-492 views

On Stochastic Models of Service System with Dependent Process Characteristics

Kosarevskaya E.S.

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.

Journal of Mathematical Sciences. 2016;214(4):493-512
pages 493-512 views

On Absolute Convergence of Series of Random Variables Almost Surely

Petrov V.V.

Abstract

New conditions are found for the absolute convergence of series of random variables almost surely. The results contain no independence assumptions. A generalization in analytical terms is obtained. Bibliography: 5 titles.

Journal of Mathematical Sciences. 2016;214(4):513-516
pages 513-516 views

Nonprobabilistic Infinitely Divisible Distributions: The Lévy-Khinchin Representation, Limit Theorems

Platonova M.V.

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

Journal of Mathematical Sciences. 2016;214(4):517-539
pages 517-539 views

Probabilities of Small Deviations of the Weighted Sum of Independent Random Variables with Common Distribution That Decreases at Zero Not Faster Than a Power

Rozovsky L.V.

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.

Journal of Mathematical Sciences. 2016;214(4):540-545
pages 540-545 views

The Mackenhoupt Condition and an Estimating Problem

Solev V.N.

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

Journal of Mathematical Sciences. 2016;214(4):546-553
pages 546-553 views

Lattice Point Problem and Questions of Estimation and Detection of Smooth Multivariate Functions

Suslina I.A.

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.

Journal of Mathematical Sciences. 2016;214(4):554-561
pages 554-561 views

Final Distribution of a Diffusion Process with Final Stop

Harlamov B.P.

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.

Journal of Mathematical Sciences. 2016;214(4):562-583
pages 562-583 views

On an Approximation for the Solutions of Some Evolution Equations by the Expectations of Random Walks Functionals

Tsykin S.V.

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

Journal of Mathematical Sciences. 2016;214(4):584-591
pages 584-591 views