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Theory of Probability and its Applications

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2000

Volume 44, Issue 4, pp. 633-807


A Finitely Additive Version of the Law of the Iterated Logarithm

I. Epifani and A. Lijoi

Theory Probab. Appl. 44, pp. 633-649 (17 pages)

Online Publication Date: July 25, 2006

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A finitely additive version of the law of the iterated logarithm (LIL) is proposed. The formulation involves only finite-dimensional distributions of a sequence of independent random variables $(X_n)_{n\ge 1}$. It is also proved that in the case where one deals with $\sigma$-additive probabilities, the given result is equivalent to the classical version of the LIL.

Entropy Numbers of Some Ergodic Averages

C. Gamet and M. Weber

Theory Probab. Appl. 44, pp. 650-668 (19 pages) | Cited 2 times

Online Publication Date: July 25, 2006

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In this paper we extend a recent remarkable covering numbers estimate for averages of contractions in a Hilbert space $H$ due to Talagrand to some moving averages of contractions. By introducing a second regularization in Talagrand's spectral regularization, we find mild conditions on the spectral measure associated to any $x\in H$, allowing estimation of the number of Hilbertian balls of radius $0 < \varepsilon\le 1$, enough to cover the subset of $H$ defined by $\{ B_n(x)=n^{-1} \sum_{j=n^2}^{n^2+n-1}U^{j}x$, $n\in \cal N\}$, where $U$ is a contraction of $H$ and $\cal N$ a geometric sequence. Moreover, we show that these conditions on the spectral measure ensure the existence of the modulus of continuity of $\{ T^{-1} \int_0^T f \circ U_t \, dt$, ${T\ge 1}\}$, where $f$ is a contraction of $L^2(\mu )$ and $\{U_t,\ t\in \bf R \}$ is a flow which preserves the measure $\mu $. Finally, we give a covering numbers estimate in a non-Hilbertian case.

Diffusion Approximation and Optimal Stochastic Control

R. Lipster, W. J. Runggaldier, and M. Taksar

Theory Probab. Appl. 44, pp. 669-698 (30 pages)

Online Publication Date: July 25, 2006

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In this paper a stochastic control model is studied that admits a diffusion approximation. In the prelimit model the disturbances are given by noise processes of various types: additive stationary noise, rapidly oscillating processes, and discontinuous processes with large intensity for jumps of small size. We show that a feedback control that satisfies a Lipschitz condition and is $\delta$-optimal for the limit model remains $\delta$-optimal also in the prelimit model. The method of proof uses the technique of weak convergence of stochastic processes. The result that is obtained extends a previous work by the authors, where the limit model is deterministic.

Stationary State and Diffusion for a Charged Particle in a One-Dimensional Medium with Lifetimes

A. Pellegrinotti, V. Sidoravicius, and M. E. Vares

Theory Probab. Appl. 44, pp. 697-721 (25 pages)

Online Publication Date: July 25, 2006

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We study a one-dimensional semi-infinite system of identical particles with random lifetimes, interacting with a charged particle (the leftmost) which is driven by a constant positive force~$F$. Particles interact through elastic collisions and at the initial time all particles are at rest, and the interparticle distances are independent identically distributed positive~random variables. Each neutral particle has an exponentially distributed lifetime, which starts counting as soon as the particle moves, and which is independent and identically distributed. Under suitable conditions we prove a strong cluster property, convergence to a limiting measure for the law of the system as seen from a charged particle, and a central limit theorem for the motion of the charged particle.

Asymptotic Efficiency of Inverse Estimators

A. C. M. van Rooij, F. H. Ruymgaart, and W. R. van Zwet

Theory Probab. Appl. 44, pp. 722-738 (17 pages)

Online Publication Date: July 25, 2006

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Inverse estimation concerns the recovery of an unknown input signal from blurred observations on a known transformation of that signal. The estimators considered in this paper are based on a regularized inverse of the transformation involved, employing a Hilbert space set-up. We focus on properties related to weak convergence. It is shown that linear functionals can be efficiently estimated in the H\'{a}jek--LeCam sense, provided they remain restricted to a suitable class. Outside this class, rates different from $\sqrt{n}$ are possible. By way of an example we present the ``convolution theorem' for a deconvolution.

Towards the Theory of Minimax-Bayesian Estimation

V. N. Soloviov

Theory Probab. Appl. 44, pp. 739-754 (16 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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The paper obtains generalizations of a theorem on normal correlation stating the linearity of minimax-Bayesian estimates and the normality of least favorable distributions. We give applications to the estimation of parameters in linear observations models.

On the 70th Birthday of Vitautas Antanovich Statulevicius

Theory Probab. Appl. 44, pp. 755-755 (1 page)

Online Publication Date: July 25, 2006

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Abstract Unavailable

New Characterization of Discrete Distributions Through Weak Records

F. A. Aliev

Theory Probab. Appl. 44, pp. 756-761 (6 pages)

Online Publication Date: July 25, 2006

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Let $X_{1},X_{2},\ldots$ be a sequence of independent and identically distributed random variables taking on values $0,1,\ldots$ with a distribution function $F$ such that $F(n) < 1$ for any $n=0,1,\ldots$ and $\mathbf{E} X_{1}\log (1+X_{1}) < \infty $. Let $X_{L(n)}$ be the $n$th weak record value and $\{ A_{k}\}_{k=0}^{\infty }$ be any sequence of positive numbers, such that $A_{k+1} > A_{k}-1$. This paper shows that if there exists an~$F(x)$, with $\mathbf{E} \{X_{L(n+2)}-X_{L(n)}\,|\, X_{L(n)}=s\}=A_{s}$ for some $n > 0$ and all $s\ge 0$, then $F(x)$ is unique.

Exit Laws and Excessive Functions for Superprocesses

E. B. Dynkin

Theory Probab. Appl. 44, pp. 762-767 (6 pages)

Online Publication Date: July 25, 2006

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Let $\xi$ be a Markov process with transition function $p(r,x;\,t,\,dy)$ and let $X$ be the corresponding Dawson--Watanabe superprocess (i.e., the superprocess with the branching characteristic $\psi(u)=\gamma u^2$). Denote by $\cal P$ the transition function of $X$ and put $$ p_n(r,x;\,t,\,dy)=\prod_{i=1}^n p(r,x_i;\,t,\,dy_i). $$ To every $p_n$-exit law $\ell$ there corresponds a $\cal P$-exit law $L_\ell$ such that, for every $t$, $L_\ell^t(\mu)$ is a polynomial of degree $n$ in $\mu$ with the leading term $\langle \ell^t,\mu^n\rangle $. Every polynomial $\cal P$-exit law has a unique representation of the form $L_{\ell_1}+\cdots+L_{\ell_n}$, where $\ell_k$ is a $p_k$-exit law.

On Strong Attraction of Stationary Sequences to a Normal Law

A. G. Grin'

Theory Probab. Appl. 44, pp. 768-775 (8 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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For stationary sequences satisfying the uniformly strong mixing condition, general conditions of attraction to the normal law are obtained in terms of distributions of particular summands. The main result of this paper generalizes all known results of this kind in the field.

On the Law of the Iterated Logarithm in Banach Lattices

I. K. Matsak

Theory Probab. Appl. 44, pp. 775-784 (10 pages)

Online Publication Date: July 25, 2006

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The law of the iterated logarithm in the classical form $$ \limsup_{n\to\infty}\frac{X_1+X_2+\cdots+X_n}{(2n\log\log(n))^{1/2}}=\mathfrak{G} X $$ is established for some Banach lattices.

On the Accuracy of Normal Approximation for the Densities of Sums of Independent Identically Distributed Random Variables

Yu. V. Zhukov

Theory Probab. Appl. 44, pp. 785-793 (9 pages) | Cited 1 time

Online Publication Date: July 25, 2006

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The structure of the nonuniform estimate of convergence rate in the local central limit theorem for the densities of sums of independent identically distributed random variables is made more accurate. The absolute constants are written out explicitly.

Isomorphisms Between Sets of Probability Distributions

A. M. Zubkov

Theory Probab. Appl. 44, pp. 794-797 (4 pages)

Online Publication Date: July 25, 2006

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Some methods permitting the construction of isomorphisms between semigroups of probability distributions (with convolution as group operation) are described, in particular, between the set of all probability distributions on a nonnegative axis and a subset of distributions on nonnegative integers. Analogous isomorphisms for the vector case and for sets of distributions corresponding to nondecreasing processes with independent increments are also described.

Reviews and Bibliography

Theory Probab. Appl. 44, pp. 798-805 (8 pages)

Online Publication Date: July 25, 2006

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Abstract Unavailable

News of Scientific Life

Theory Probab. Appl. 44, pp. 806-807 (2 pages)

Online Publication Date: July 25, 2006

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Abstract Unavailable
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