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Theory of Probability and its Applications

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1994

Volume 39, Issue 4, pp. 537-726


Asymptotically Ergodic Markov Functionals of an Ergodic Process

D. Alimov

Theory Probab. Appl. 39, pp. 537-546 (10 pages)

Online Publication Date: July 17, 2006

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Let $X(t)$ be a homogeneous Markov process given on a state space $(E,\mathcal{B})$ and having an invariant distribution $\pi ( \cdot )$. Let $\{ \xi _n (t)\} $ be a sequence of cut-off Markov functionals with killing times $\{ \xi _n \} $ and a set of values $I = \{ 1,2, \ldots ,d\} $ which converges to a trivial functional with a stationary distribution $\rho ( \cdot )$. We give the conditions under which there exists a sequence $\varepsilon _n \to + 0$ such that if the inequality ${\bf P}_{\pi ,\rho } \{ \xi _n < \infty \} > 0$ holds for all sufficiently large $n$, then for any $t \geqq 0,x \in E,i,j \in I$, and all continuous bounded functions $\varphi (y),y \in E$, \[ \mathop {\lim }\limits_{n \to \infty } {\bf P}_{x,i} \left[ {\varphi \left( {X\left( {\tfrac{t}{{\varepsilon _n }}} \right)} \right),\xi _n \left( {\frac{t}{{\varepsilon _n }}} \right) = j} \right] = e^{ - t} \rho (j)\int_E {\pi (dy)\varphi (y).} \]

On the Problem of Detecting Random Trajectories

I. M. Arbekov, V. I. Vinokurov, and B. V. Ryazanov

Theory Probab. Appl. 39, pp. 547-557 (11 pages)

Online Publication Date: July 17, 2006

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A consistent test is constructed for the hypothesis that there is a particle which performs random walk along the integer lattice of the real line and carries a “useful signal” in the presence of “noise.”. The limiting form of the linear approximation to the likelihood ratio is found to be a convolution of the standard normal distribution and a functional of the standard Wiener process.

Martingale Methods for Random Walks in a One-Dimensional Random Environment

A. A. Butov

Theory Probab. Appl. 39, pp. 558-572 (15 pages)

Online Publication Date: July 17, 2006

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One-dimensional random walk processes in a random environment of a general functional type are considered. The study is carried out by the natural scale method. We obtain conditions of existence of the natural scale, conditions of existence of the processes and a theorem on the representation of the local time as the compensator of the modulus of the martingale which is the random walk in the natural scale. The work is performed in martingale terms and contains a number of examples.

Itô Formula for an Extended Stochastic Integral with Nonanticipating Kernel

N. V. Norin

Theory Probab. Appl. 39, pp. 573-592 (20 pages)

Online Publication Date: July 17, 2006

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Let $U_t = \smallint _0^1 u_s \mu (t,s)\delta W_s $ be an extended stochastic integral with a nonrandom anticipating kernel $\mu ( \cdot , \cdot )$. This paper gives the conditions of continuity for the process $U_t $ (§ 3), computes the quadratic variation (§ 4), and proves the Itô formula (§ 5) from which the formula for Brownian partial derivatives is deduced. With the help of the established Ito formula the probabilistic solution of some integro-differential equation is obtained (Example 3).

Poisson Approximation for the Number of Long Match Patterns in Random Sequences

Yu. S. Novak

Theory Probab. Appl. 39, pp. 593-603 (11 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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Let $X_1 , \ldots ,X_m ,Y_1 , \ldots ,Y_n $ be independent identically distributed random variables with discrete state space. We estimate the rate of convergence in the limit theorems for the number of long match patterns and for the length of the longest match pattern in random sequences $X_1 , \ldots ,X_m ,Y_1 , \ldots ,Y_n $. The results improve the corresponding ones received by Zubkov–Mikhailov, Arratia–Gordon–Waterman, and others.

A Storage Model for Data Communication Systems

N. U. Prabhu and A. Pacheco

Theory Probab. Appl. 39, pp. 604-627 (24 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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We consider a storage model where the input and demand are modulated by an underlying Markov chain. Such models arise in data communication systems. The input is a Markov-compound Poisson process and the demand is a Markov linear process. The demand is satisfied if physically possible. We study the properties of the demand and its inverse, which may be viewed as transformed time clocks. We show that the unsatisfied demand is related to the infimum of the net input and that, under suitable conditions, it is an additive functional of the input process. The study of the storage level is based on a detailed analysis of the busy period, using techniques based on infinitesimal generators. The Laplace transform of the busy period is the unique solution of a certain matrix-functional equation. Steady state results are also obtained; these are not obvious generalizations of the results for simple storage models. In particular, a generalization of the Pollaczek-Khinchin formula brings new insight.

Branching Processes with Final Types of Particles and Random Trees

V. A. Vatutin

Theory Probab. Appl. 39, pp. 628-641 (14 pages)

Online Publication Date: July 17, 2006

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This paper considers a Bellman–Harris branching process whose probability generating function $f(s)$ of the number of direct descendants of particles satisfies the relation $f(s) = s + (1 - s)^{1 + \alpha } L(1 - s),0 < \alpha \leqq 1$. Let $\tau $ be the moment of extinction of the process and let $\nu_\Delta $ be the total number of particles the number of direct descendants of each of which belongs to the set $\Delta ,\Delta \subset \{ 0,1, \ldots ,n, \ldots \} $. The paper gives conditions under which, for any $x \in ( - \infty , + \infty )$ and some scaling constants $b(N)$, a nondegenerate limit, $\lim _{N \to \infty } {\bf P}\{ \tau b(N) \leqq x|\nu_\Delta = N\} $, exists.

Dual Processes and Ergodic Type Theorems for Markov Chains in the Scheme of Series

Yu. A. Velikij, A. I. Motsa, and D. S. Sil’vestrov

Theory Probab. Appl. 39, pp. 642-653 (12 pages)

Online Publication Date: July 17, 2006

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Resolvent type necessary and sufficient conditions of weak convergence and convergence of moments are investigated in the scheme of series, for non-negative additive functionals over Markov chains with an arbitrary phase space.

E. B. Dynkin’s 70th Birthday

Theory Probab. Appl. 39, pp. 654-656 (3 pages)

Online Publication Date: July 17, 2006

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Abstract Unavailable

On Refinement of Banach-Valued Limit Theorems for Stable Laws

A. N. Chuprunov

Theory Probab. Appl. 39, pp. 657-662 (6 pages)

Online Publication Date: July 17, 2006

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This paper gives the estimates of rate of convergence on classes of sets in Banach-valued limit theorems for stable laws. These estimates are effective in the case when the limit distribution is concentrated on a space of lower dimension and have unimprovable order.

Distributions of Itô Processes: Estimates for the Density and for Conditional Expectations of Integral Functionals

N. G. Dokuchaev

Theory Probab. Appl. 39, pp. 662-670 (9 pages) | Cited 4 times

Online Publication Date: July 17, 2006

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We obtain a priori estimates for the $L_2 $-norms of solutions of parabolic Itô equations describing evolution of the distributions of solutions of ordinary Itô stochastic differential equations with random coefficients. In the case of nondegenerate equations, estimates for the $L_2 $-norms of derivatives with respect to space variables are also obtained. As a consequence, we establish a generalisation of Itô’s formula for functions that have only square-summable derivatives of the first and the second order (or even of the first order).

Integral Transforms with Infinitely Divisible Kernels

M. Finkelstein, S. Scheiberg, and H. G. Tucker

Theory Probab. Appl. 39, pp. 670-676 (7 pages)

Online Publication Date: July 17, 2006

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Given $r$ characteristic functions $f_1 (u), \ldots ,f_r (u)$, none of which is identically equal to one, it is shown that the integral transform\[ \int_0^\infty \cdots \int_0^\infty {\left( {\prod\limits_{j = 1}^r {fj(u_j )^{s_j } } } \right)dF(s_1 , \ldots ,s_r )} \] of the joint distribution function $F$ of $r$ non-negative random variables can be defined over a nonempty domain of natural numbers and it uniquely determines $F$. This result is used to obtain the converse of a multivariate version of a transfer theorem due to Gnedenko and Fahim, thus extending a result of Szasz and Frajeris in the univariate case. An application is also made to Lévy processes.

On the Strong Law of Large Numbers for Blockwise Independent and Blockwise Orthogonal Random Variables

V. F. Gaposhkin

Theory Probab. Appl. 39, pp. 677-684 (8 pages) | Cited 3 times

Online Publication Date: July 17, 2006

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In this paper we obtain the conditions of the strong law of large numbers for systems of random variables which are blockwise independent and blockwise orthogonal. We show that these conditions are unimprovable for the classes of random variables considered.

Canonical Spectral Equation

V. L. Girko

Theory Probab. Appl. 39, pp. 685-691 (7 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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We consider a sequence of symmetric real-valued random matrices $\Xi _n = (\xi _{ij}^{(n)} )_{i,j = 1}^n ,n = 1,2, \ldots $, whose entries $\xi _{ij}^{(n)} ,i \geqq j,i,j = 1, \ldots ,n,$, are independent for each $n$, whereas ${\bf E}\xi _{ij}^{(n)} = a_{ij}^{(n)} ,{\text{Var}}\xi _{ij}^{(n)} = \sigma _{ij}^{(n)} ,i \geqq j,i,j = 1, \ldots ,n,$\[ \mathop {\sup }\limits_n \mathop {\max }\limits_{i = 1, \ldots ,n} \sum\limits_{j = 1}^n {\sigma _{ij}^{(n)} < \infty} ,\qquad \mathop {\sup }\limits_n \mathop {\max }\limits_{i = 1, \ldots ,n} \sum\limits_{j = 1}^n {\left| {a_{ij}^{(n)} } \right| < \infty ,} \]and the Lindeberg condition is satisfied for these entries: for any $\tau > 0$,\[ \mathop {\lim }\limits_{n \to \infty } \mathop {\max }\limits_{i = 1, \ldots ,n} \sum\limits_{j = 1}^n {{\bf E}\left[ {\xi _{ij}^{(n)} - a_{ij}^{(n)} } \right]^2 \chi \left\{ {|\xi _{ij}^{(n)} - a_{ij}^{(n)} | > \tau } \right\} = 0.} \] We prove that $p\lim _{n \to \infty } \sup _x |\mu _n (x) - F_n (x)| = 0$, where $\mu _n (x) = n^{ - 1} \Sigma _{k = 1}^n \chi (\omega :\lambda _k < x),\lambda _1 \geqq \cdots \geqq \lambda _n $ are the eigenvalues of the random matrix $\Xi _n = (\xi _{ij}^{(n)} )_{i,j = 1}^n ,F_n (x)$ are distribution functions, the Stieltjes transforms of which are equal to \[\int {(x - z)^{ - 1} dF_n (x) = n^{ - 1} \sum\limits_{i = 1}^n {c_i (z),\quad z = t + is,\quad s \ne 0,} } \]and the functions $c_i (z)$ satisfy the system of equations\[ c_i (z) = \left\{ {\left[ {A - zI_n - \delta _{pl} \sum\limits_{s = 1}^n {c_s (z)\sigma _{sl}^{(n)} } } \right]^{ - 1} } \right\}_{ii} ,\quad i = 1, \ldots ,n, \] where $\delta _{pl} $ is the Kronecker symbol, $A_n = (a_{ij}^{(n)} )_{i,j = 1}^n ,I_n $ is the identity matrix of the $n$th order.

Interoutput Times in an $M/G/1/PS$ Queue with Instant Feedback

S. A. Grishechkin

Theory Probab. Appl. 39, pp. 692-696 (5 pages)

Online Publication Date: July 17, 2006

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We consider an $M/G/1/PS$ system in which customers can immediately rejoin the queue upon completion of the service. The Laplace transform is found for the time interval between two successive epochs the server becomes free (in the stationary regime).

On Estimation of Maxima of Sums of Random Variables Indexed by Edges of Graphs

F. I. Karpelevich

Theory Probab. Appl. 39, pp. 696-702 (7 pages)

Online Publication Date: July 17, 2006

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This paper considers a family of independent identically distributed random variables that are indexed by the edges of a graph. The maximum of sums of such variables along the paths of the graph is studied. We show that if one graph covers another one, then the maximum of sums for the first graph is stochastically greater than that for the second graph.

Some Problems of Statistics of Distribution Mixtures with Variable Concentrations

R. E. Maiboroda

Theory Probab. Appl. 39, pp. 703-707 (5 pages)

Online Publication Date: July 17, 2006

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Some problems of estimating characteristics of random variables are considered. The observations used as a basis are a sample from a mixture of distributions, and the concentrations of components vary in the process of measurement. The distributions of components of the mixture are evaluated nonparametrically, while estimates for some functions of concentrations are both parametric and nonparametric. The estimates considered are proved to be consistent. Estimates are constructed on the basis of inhomogeneous empirical distribution functions.

Two Limit Theorems for Diffusion Type Stochastic Equations

S. I. Pisanets

Theory Probab. Appl. 39, pp. 708-713 (6 pages)

Online Publication Date: July 17, 2006

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This paper investigates the convergence conditions of solutions of diffusion type stochastic equations to the solution of the same type equation.

On Certain Asymptotic Properties of Waiting Time in a Multiserver Queueing System with Identical Times

O. P. Vinogradov

Theory Probab. Appl. 39, pp. 714-718 (5 pages) | Cited 1 time

Online Publication Date: July 17, 2006

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A multiphase queueing system is considered. The service time of the nth arrival at the $i$th server is $T_{n,i} $ and ${\bf P}\{ T_{n,1} = \cdots = T_n \} = 1$, where $\{ T_n \} $ are independent identically distributed random variables with an arbitrary common distribution.
Let $U_l (n)$ be the time spent by the $l$ arrival at the $l$th server. Some algebraic properties of the sequence $\{ U_l (n)\} (l \geq 2)$ are cleared up. In the case of Poisson input flow, the distributions of some characteristics of the system are obtained, as well as a number of limit theorems for the situation where the number of servers grows infinitely.

Addendum: Asymptotic Expansions in the Problem of Sequential Estimation of an Autoregressive Parameter

V. K. Malinovskii

Theory Probab. Appl. 39, pp. 719-719 (1 page)

Online Publication Date: July 17, 2006

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Abstract Unavailable

Reviews amd Bibliography

Theory Probab. Appl. 39, pp. 720-725 (6 pages)

Online Publication Date: July 17, 2006

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Abstract Unavailable

New Books

Theory Probab. Appl. 39, pp. 726-726 (1 page)

Online Publication Date: July 17, 2006

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Abstract Unavailable
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