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April 2012
The 20 articles with the most full-text downloads during the month, in descending order.
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Theory Probab. Appl. 56, pp. 21-43 (23 pages)
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We obtain analogues of the well-known Chebyshev's exponential inequality ${\bf P}(\xi \ge x)\le e^{-\Lambda^{(\xi)}(x)}$, $x>{\bf E}\,\xi,$ for the distribution of a random variable $\xi$, where $\Lambda^{(\xi)}(x):=\sup_\lambda\{\lambda x- \log {\bf E}\,e^{\lambda \xi}\}$ is the large deviation rate function for $\xi$. Generalizations of this relation are established for multivariate random vectors $\xi$, for sums of the vectors, and for trajectories of random processes associated with such sums. |
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On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities Theory Probab. Appl. 16, pp. 264-280 (17 pages) Online Publication Date: July 17, 2006
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Theory Probab. Appl. 56, pp. 1-20 (20 pages)
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In the first part of this paper we give an elementary proof of the fact that if an infinite matrix $A$, which is invertible as a bounded operator on $\ell^2$, can be uniformly approximated by banded matrices, then so can the inverse of $A$. We give explicit formulas for the banded approximations of $A^{-1}$ as well as bounds on their accuracy and speed of convergence in terms of their bandwidth. We then use these results to prove that the so-called Wiener algebra is inverse closed. In the second part of the paper we apply these results to covariance matrices $\Sigma$ of Gaussian processes and study mixing and beta mixing of processes in terms of properties of $\Sigma$. Finally, we note some applications of our results to statistics. |
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Non-Parametric Estimation of a Multivariate Probability Density Theory Probab. Appl. 14, pp. 153-158 (6 pages) Online Publication Date: July 17, 2006
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Central Limit Theorem for Certain Classes of Dependent Random Variables Theory Probab. Appl. 51, pp. 335-342 (8 pages) Online Publication Date:
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Our main interest in this article is the central limit theorem (CLT) for dependent classes of random variables. We computed the exact difference between the characteristic function of the sum of dependent random variables and the characteristic function of the sum of independent random variables of the same distributions. The obtained difference serves for defining the class of dependent random variables such that usual convergence in distribution to the normally distributed random variable holds. The dependency structure, as defined in our class of random variables, can be reflected in some physical phenomena. |
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Uncertain Change-Point Problem for Stochastic Sequence Theory Probab. Appl. 56, pp. 44-56 (13 pages)
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The change-point problem for a sequence of independent random variables is considered. Distributions of random variables before and after a change-point are unknown, but a finite collection of possible distributions is known a priori. Therefore, the problem is to detect the change without any information about its direction (an “uncertain change-point problem”). A new vector criterion to be minimized is proposed for change-point detection method quality estimation. For this criterion, nonasymptotic lower bounds are obtained. A method of quickest detection of the uncertain change-point is proposed for which these lower bounds are asymptotically attained. |
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On a Statistical Estimate for the Entropy of a Sequence of Independent Random Variables Theory Probab. Appl. 4, pp. 333-336 (4 pages) Online Publication Date: July 17, 2006
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The mean value and variance are computed for a statistical estimate for the entropy of a sequence of mutually independent random variables having a similar distribution. The estimate is shown to be biased, consistent and asymptotically normal. |
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Final Probabilities for a Branching Process with Interaction of Particles and an Epidemic Process Theory Probab. Appl. 43, pp. 633-640 (8 pages) Online Publication Date: July 25, 2006
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The exponential generating function method suggested in [A. V. Kalinkin, Extinction probability of a branching process with interaction of particles, Theory Probab. Appl., 27 (1982), pp.~201--205] and [A.~V.~Kalinkin, Final probabilities for a branching random process with interaction of particles, Dokl. Akad. Nauk SSSR, 269 (1983), pp.~1309--1312 (in Russian)] to solve the stationary first (backward) Kolmogorov system of differential equations is applied to the Weiss epidemic model and its generalizations. Integral representations for the generating functions of the final probabilities are obtained. |
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In memory of Anatolii Vladimirovich Skorokhod (1930–2011) Theory Probab. Appl. 56, pp. 116-119 (4 pages)
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A review of the career of Anatolii Vladimirovich Skorokhod, who passed away on January 4, 2011. He was an outstanding scientist, the author of fundamental works in the theory of probability and statistics, and academician of the National Ukrainian Academy of Sciences. |
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Theory Probab. Appl. 9, pp. 141-142 (2 pages) Online Publication Date: July 17, 2006
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A study is made of certain properties of an approximation to the regression line on the basis of sampling data when the sample size increases unboundedly. |
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Convergence of Random Processes and Limit Theorems in Probability Theory Theory Probab. Appl. 1, pp. 157-214 (58 pages) Online Publication Date: July 28, 2006
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The convergence of stochastic processes is defined in terms of the so-called “weak convergence” (w. c.) of probability measures in appropriate functional spaces (c. s. m. s.). Chapter 1. Let $\Re $ be the c.s.m.s. and $v$ a set of all finite measures on $\Re $. The distance $L(\mu _1 ,\mu _2 )$ (that is analogous to the Lévy distance) is introduced, and equivalence of $L$-convergence and w. c. is proved. It is shown that $V\Re = (v,L)$ is c. s. m. s. Then, the necessary and sufficient conditions for compactness in $V\Re $ are given. In section 1.6 the concept of “characteristic functionals” is applied to the study of w. cc of measures in Hilbert space. Chapter 2. On the basis of the above results the necessary and sufficient compactness conditions for families of probability measures in spaces $C[0,1]$ and $D[0,1]$ (space of functions that are continuous in $[0,1]$ except for jumps) are formulated. Chapter 3. The general form of the “invariance principle” for the sums of independent random variables is developed. Chapter 4. An estimate of the remainder term in the well-known Kolmogorov theorem is given (cf. [3.1]). |
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On the Upper Bound for the Absolute Constant in the Berry–Esseen Inequality Theory Probab. Appl. 54, pp. 638-658 (21 pages) Online Publication Date: November 11, 2010
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This paper describes the history of the search for unconditional and conditional upper bounds of the absolute constant in the Berry–Esseen inequality for sums of independent identically distributed random variables. Computational procedures are described. New estimates are presented from which it follows that the absolute constant in the classical Berry–Esseen inequality does not exceed 0.5129. |
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Theory Probab. Appl. 13, pp. 197-224 (28 pages) Online Publication Date: July 28, 2006
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Some Limit Theorems for Stationary Processes Theory Probab. Appl. 7, pp. 349-382 (34 pages) Online Publication Date: July 17, 2006
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In this paper stationary stochastic processes in the strong sense are investigated, which satisfy the condition \[ | {{\bf P} ( {AB} ) - {\bf P} ( A ){\bf P} ( B )} | \leqq \varphi ( n ){\bf P}( A ),\quad \varphi ( n ) \downarrow 0, \] for every $A \in \mathfrak{M}_{ - \infty }^0 ,B \in \mathfrak{M}_n^\infty $, or the “strong mixing condition” \[ \mathop {\sup }\limits_{A \in \mathfrak{M}_{ - \infty }^0 ,B \in \mathfrak{M}_n^\infty } | {{\bf P} ( {AB} ) - {\bf P} ( A ){\bf P} ( B )} |\alpha ( n ) \downarrow 0, \] where $\mathfrak{M}_a^b $ is a $\sigma $-algebra generated by the events \[ \{ {( {x_{i_1 } ,x_{i_2 } , \cdots ,x_{i_k } } ) \in {\bf E}} \},\qquad a \leqq i_1 < i_2 < \cdots < i_k \leqq b, \]${\bf E}$ being a $k$-dimensional Borel set. Some limit theorems for the sums of the type \[ \frac{{x_1 + \cdots + x_n }}{{B_n }} - A_n \quad {\text{or}} \quad \frac{{f_1 + \cdots + f_n }}{{B_n }} - A_n \] are established. Here $f_j = T^j f$, and the random variable $f$ is measurable with respect to $\mathfrak{M}_{ - \infty }^\infty $. |
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Recurrence Relations for Price Bounds of Contingent Claims in Discrete Time Market Models Theory Probab. Appl. 56, pp. 72-95 (24 pages)
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We reduce the computation of price bounds of contingent claims to a sequence of interconnected finite dimensional optimization problems, depending on a parameter. In a perfect market model the obtained formulas characterize upper hedging price, while in a multicurrency market model with transaction costs they describe the set of initial portfolios, allowing for superhedging of a vector contingent claim. The mentioned formulas do not contain martingale measures or their analogues. The proofs are based on the martingale selection theorem. The effectiveness of the proposed approach is illustrated by several examples. |
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Large Deviations of the Empirical Current in Interacting Particle Systems Theory Probab. Appl. 51, pp. 2-27 (26 pages) Online Publication Date: February 22, 2007
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We study current fluctuations in lattice gases in the hydrodynamic scaling limit. More precisely, we prove a large deviation principle for the empirical current in the symmetric simple exclusion process with rate functional $I$. We then estimate the asymptotic probability of a fluctuation of the average current over a large time interval and show that the corresponding rate function can be obtained by solving a variational problem for the functional $I$. For the symmetric simple exclusion process the minimizer is time independent so that this variational problem can be reduced to a time‐independent one. On the other hand, for other models the minimizer is time dependent. This phenomenon is naturally interpreted as a dynamical phase transition. |
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The Monge–Kantorovich Mass Transference Problem and Its Stochastic Applications Theory Probab. Appl. 29, pp. 647-676 (30 pages) Online Publication Date: July 17, 2006
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Theory Probab. Appl. 26, pp. 803-806 (4 pages) Online Publication Date: July 17, 2006
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Mixing Conditions for Markov Chains Theory Probab. Appl. 18, pp. 312-328 (17 pages) Online Publication Date: July 17, 2006
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A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides Theory Probab. Appl. 11, pp. 390-406 (17 pages) Online Publication Date: July 17, 2006
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