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2005

Volume 49, Issue 4, pp. 561-744


On Sharp Large Deviations for Sums of Random Vectors and Multidimensional Laplace Approximation

Ph. Barbe and M. Broniatowski

Theory Probab. Appl. 49, pp. 561-588 (28 pages)

Online Publication Date: July 25, 2006

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Let $X, X_i,i\ge 1$, be a sequence of independent and identically distributed random vectors in ${\bf R}^d$. Consider the partial sum $S_n:=X_1+\cdots +X_n$. Under some regularity conditions on the distribution of $X$, we obtain an asymptotic formula for $P\{S_n\in nA\}$, where $A$ is an arbitrary Borel set. Several corollaries follow, one of which asserts that, under the same regularity conditions, for any Borel set $A$, $\lim_{n\to\infty}n^{-1}\log P\{S_n\in nA\} =-I(A)$, where $I$ is a large deviation functional. We also prove a multidimensional Laplace-type approximation that allows an explicit calculation of the sharp large deviation probability typically when the set $A$ has a smooth boundary.

A Probabilistic Approach to a Solution of Nonlinear Parabolic Equations

Ya. Belopolskaya

Theory Probab. Appl. 49, pp. 589-611 (23 pages)

Online Publication Date: July 25, 2006

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We construct a probabilistic representation of the Cauchy problem solution for a system of nonlinear parabolic equations and give the conditions which guarantee that this representation can be applied to construct and investigate a solution of the Cauchy problem for a system of nonlinear hyperbolic equations. As an example, we consider the system of gas dynamic equations and its parabolic regularization.

On the Central Limit Theorem for Toeplitz Quadratic Forms of Stationary Sequences

M. S. Ginovyan and A. A. Sahakyan

Theory Probab. Appl. 49, pp. 612-628 (17 pages) | Cited 6 times

Online Publication Date: July 25, 2006

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Let $X(t)$, $t = 0,\pm1,\ldots$, be a real-valued stationary Gaussian sequence with a spectral density function $f(\lambda)$. The paper considers the question of applicability of the central limit theorem (CLT) for a Toeplitz-type quadratic form $Q_n$ in variables $X(t)$, generated by an integrable even function $g(\lambda)$. Assuming that $f(\lambda)$ and $g(\lambda)$ are regularly varying at $\lambda=0$ of orders $\alpha$ and $\beta$, respectively, we prove the CLT for the standard normalized quadratic form $Q_n$ in a critical case $\alpha+\beta={\frac{1}{2}}$.
We also show that the CLT is not valid under the single condition that the asymptotic variance of $Q_n$ is separated from zero and infinity.

Poisson Approximation of Increment Processes with Markov Switching

V. S. Korolyuk and N. Limnios

Theory Probab. Appl. 49, pp. 629-644 (16 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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In this paper we present weak convergence results for Markov switched increment processes toward a compound Poisson process. The switching Markov process is considered also with an asymptotic split phase space. These results are obtained by a semimartingale approach.

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

A. I. Nazarov and Ya. Yu. Nikitin

Theory Probab. Appl. 49, pp. 645-658 (14 pages)

Online Publication Date: July 25, 2006

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We find the logarithmic L2-small ball asymptotics of some Gaussian processes related to the fractional Brownian motion (fBm), fractional Ornstein--Uhlenbeck process (fOU), and their integrated analogues. We consider also the multiparameter generalizations.

Limit Theorems for Allocation of Particles over Different Cells with Restrictions to the Size of the Cells

A. N. Timashev

Theory Probab. Appl. 49, pp. 659-670 (12 pages) | Cited 3 times

Online Publication Date: July 25, 2006

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Equiprobable allocation schemes of $n$ indistinguishable or distinguishable particles over $N$ distinguishable cells are considered provided the fillings of the cells take on values in a fixed subset $A$ of the set of nonnegative integers. Local normal and Poisson theorems are proved for the distributions of the number of cells, each of which contains exactly $r$ particles, and for the number of cycles of length $r\in A$ in a permutation selected at random and equiprobable from the set of all permutations of order $n$ with $N$ cycles $(N\le n)$ whose lengths are elements of a set $A\subset{\bf N}$. It is assumed that $n,N\to\infty$ in the central domain.

On Large Deviations, II

S. V. Zhulenev

Theory Probab. Appl. 49, pp. 671-690 (20 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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The Esscher--Cramér approach enables us to get representations of large deviation type even if the Cramér condition fails. Moreover, in the case of the scaled sums of a sequence of independent identically distributed random variables these representations turn out to be slightly different from those which are correct under this rigid condition.

On the 70th Birthday of A. N. Shiryaev

translated by Ya. G. Sinai

Theory Probab. Appl. 49, pp. 691-694 (4 pages)

Online Publication Date: July 25, 2006

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This paper is dedicated to the leading Russian probabilist Albert N. Shiryaev on the occasion of his 70th birthday, which took place on October 12, 2004.

Short Communications On Inequalities for Large Deviations in the Bernoulli Scheme

V. M. Kruglov

Theory Probab. Appl. 49, pp. 695-700 (6 pages)

Online Publication Date: July 25, 2006

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We discuss known inequalities for probabilities of large deviations for sums of Bernoulli random variables and improve some of them.

Maxima of Independent Sums in the Presence of Heavy Tails

A. V. Lebedev

Theory Probab. Appl. 49, pp. 700-703 (4 pages)

Online Publication Date: July 25, 2006

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Let $$Y_{mn}=\max_{1\le i\le m}\sum_{j=1}^n X_{ij},\qquad m,n\ge 1,$$ be a family of extremes, where $X_{ij}$, $i,j\ge 1$, are independent with common subexponential distribution~$F$. The limit behavior of $Y_{mn}$ is investigated as $m,n\to\infty$. Various nondegenerate limit laws are obtained (Fr\'echet and Gumbel), depending on the relative rate of growth of $m,n$ and the tail behavior of~$F$.

On Large Deviations of a Self-normalized Sum

S. V. Nagaev

Theory Probab. Appl. 49, pp. 704-713 (10 pages)

Online Publication Date: July 25, 2006

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In the present paper we deduce exponential bounds on the probabilities of large deviations of a self-normalized sum of independent random variables. Summands are not assumed to be identically distributed.

Absolute Continuity between a Gibbs Measure and Its Translate

E. Nowak

Theory Probab. Appl. 49, pp. 713-724 (12 pages)

Online Publication Date: July 25, 2006

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We look for an overestimation of the distance in total variation between a Gibbs measure on ${\bf R}^{{\bf Z}^d}$ and its translate by a vector of this space. This can be done thanks to a control of the interdependence between the spins at distinct sites, i.e., prescribing some restrictions for the associated potential. We can then conclude, for precise cases, with the equivalence of the initial measure and its translate.

On Exact Asymptotics in the Weak Law of Large Numbers for Sums of Independent Random Variables with a Common Distribution Function from the Domain of Attraction of a Stable Law. II

L. V. Rozovsky

Theory Probab. Appl. 49, pp. 724-734 (11 pages) | Cited 3 times

Online Publication Date: July 25, 2006

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Let us consider independent identically distributed random variables $X_1, X_2, \dots\,$, such that $$ U_n=\frac{S_n}{B_n} -n\,a_n \longrightarrow \xi_\alpha\qquad \mbox{weakly as}\quad n\to\infty, $$ where $S_n = X_1 + \cdots + X_n$, $B_n>0$, $a_n$ are some numbers $(n\geq 1)$, and a random variable~$\xi_\alpha$ has a stable distribution with characteristic exponent~$\alpha\in[1,2]$. Our basic purpose is to find conditions under which $$ \sum_n f_n{\bf P}\big\{U_n\geq\varepsilon\varphi_n\big\}\sim \sum_n f_n{\bf P}\big\{\xi_\alpha\ge\varepsilon\varphi_n\big\}, \qquad\varepsilon\searrow 0, $$ with a positive sequence $\varphi_n$, which tends to infinity and satisfies mild additional restrictions, and with a nonnegative sequence $f_n$ such that $\sum_n f_n =\infty $.

On the Necessary Conditions of Poisson Convergence for Martingales

A. G. Sholomitskii

Theory Probab. Appl. 49, pp. 735-737 (3 pages)

Online Publication Date: July 25, 2006

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This paper proves a theorem on necessary conditions for the Poisson convergence of sums of martingale differences. The theorem conditions generalize the classic conditions for the convergence of sums of independent summands.

A Renewal Equation in a Multidimensional Space

N. B. Yengibarian

Theory Probab. Appl. 49, pp. 737-744 (8 pages)

Online Publication Date: July 25, 2006

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The following renewal equation in a multidimensional space (REMS) is considered $$ f(x)=g(x)+\int_{{\bf R}^n}K(x-t)\,f(t)\,dt, $$ where $K$~is the density of a distribution in ${\bf R}^n$. Assuming that $g\in L_1({\bf R}^n)$ and that the nonzero vector of the first moment of $K$ is finite we prove the existence and uniqueness of a solution of an REMS within a certain class of functions. The renewal density for the solution of this equation is constructed and its properties are investigated. We give a probabilistic interpretation for our results by means of an example from the theory of random walks in~${\bf R}^n$.
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