Almost Sure Hausdorff Dimension of Graphs of Random Wavelet Series

In this contribution, we consider the problem of computing the Hausdorff dimension of the graph of a continuous random field obtained as an infinite series of smooth deterministic functions with independent random weights. The particular case of random wavelet series is addressed and almost sure lower bounds of their Hausdorff dimension are obtained. Sub-classes are exhibited for which these lower bounds coincide with almost sure upper bounds based on particular smoothness indices of the series. A direct application of these results provides new insights concerning the Hausdorff dimension as opposed to classical smoothness indices.     

A remark on the law of the logarithm for weighted sums of random variables with multidimensional indices

In this work, a sharp upper bound on the law of the logarithm for the weighted sums of random variables with multidimensional indices is obtained. The main result improves the result in [Li, Rao and Wang, 1995. On strong law of large numbers and the law of the logarithm for weighted sums of independent random variables with multidimensional indices. J. Multivariate Anal. 52, 181–198], partly.     

On the residual dependence index of elliptical distributions

The residual dependence index of bivariate Gaussian distributions is determined by the correlation coefficient. This tail index is of certain statistical importance when extremes and related rare events of bivariate samples with asymptotic independent components are being modeled. In this paper we calculate the partial residual dependence indices of a multivariate elliptical random vector assuming that the associated random radius has distribution function in the Gumbel max-domain of attraction. Furthermore, we discuss the estimation of these indices when the associated random radius possesses a Weibull-tail distribution.     

Some new limit theorems for vector space- and group-valued random variables

It is shown that weakly convergent sums (products) of normalized i.i.d. random variables with values in a finite-dimensional vector space or in a group are mixing in the sense of A. Rényi, and limit theorems for random sums with nonindependent indices are obtained. A new version of H. Robbins' limit theorem for random sums is presented. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló Hungary, 1997, Part III.     

On almost sure limiting behavior of weighted sums of random fields

We study necessary and sufficient conditions for the almost sure convergence of averages of independent random variables with multidimensional indices obtained by certain summability methods.     

Stable convergence of random sequences with random indices

In this paper, we give a complete solution to Rényi's conjecture about stable convergence under a general metric space setup, and provide new Anscombe-type theorems concerning the stable convergence of random sequences with random indices. In particular, we present a random version of the famous arcsine law. Supported by the Postdoctoral Foundation of China. Proceedings of the Seminar on Stability Problems for Stochastic Models, Vologda, Russia, 1998, Part II.     

APPROXIMATING ANALYSIS OF A KIND OF REPAIRABLE STANDBY SYSTEMS

ＬＩＷＥＩ（李伟）；ＣＡＯＪＩＮＨＵＡ（曹晋华）（ＩｎｓｔｉｔｕｔｅｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ，ｔｈｅＣｈｉｎｅｓｅＡｃａｄｅｍｙｏｆＳｃｉｅｎｃｅｓ，Ｂｅｉｊｉｎｇ１０００８０，ＣｈｉｎａａｎｄＡｓｉａ－ＰａｃｉｆｉｃＯｐｅｒａｔｉｏ...     

Convergence of multidimensional random sequences with independent random indices

The paper present necessary and sufficient conditions for the weak convergence of R^m-valued random sequences with independent random indices under some additional assumptions. Operator normalization is considered. Supported by the Russian Foundation for Fundamental Research (grant No. 93-01-01446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.     

On extremal indices greater than one for a scheme of series

The classic extremal index θ is an important parameter of asymptotic behavior of maxima of stationary random sequences. For applications, however, it is interesting to investigatemore complex structures. Recently, the extremal index was generalized to a scheme of series of random length. If the exact extremal index does not exist, then we consider partial indices. In contrast to the classic index, partial indices can be greater than one. In this paper, we consider a new model, where left and right partial indices can be greater than one and equal to each other, although the exact index does not exist.     

配备FGM Copulas二维随机向量之和的相依性

本文研究了配备Farlie-Gumbel-Morgenstern Copulas的二维随机向量之和的相依性,得到了在这类Copulas函数下两个独立的随机向量之和的Kendall及Spearman相依系数的一般公式;并针对边缘分布分别为指数分布的情况推导出了具体的公式;证明了当边缘分布满足一定的条件时,不存在尾部相依性.此外,对于几种不同边缘分布的情况进行了随机模拟与比较.这些方法及结果对两个企业(公司)合并后某两个随机指标之间的相依性问题的研究具有理论指导意义,为这类问题的进一步探索提供了理论基础.     

A limit theorem for multiple sums of identically distributed independent random variables

One investigates the asymptotic behavior (with respect to a small parameter) of series of probabilities of large deviations for sums of random variables with multidimensional indices.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 265–277, 1986.     

Random walk centrality and a partition of Kemeny’s constant

We consider an accessibility index for the states of a discrete-time, ergodic, homogeneous Markov chain on a finite state space; this index is naturally associated with the random walk centrality introduced by Noh and Reiger (2004) for a random walk on a connected graph. We observe that the vector of accessibility indices provides a partition of Kemeny’s constant for the Markov chain. We provide three characterizations of this accessibility index: one in terms of the first return time to the state in question, and two in terms of the transition matrix associated with the Markov chain. Several bounds are provided on the accessibility index in terms of the eigenvalues of the transition matrix and the stationary vector, and the bounds are shown to be tight. The behaviour of the accessibility index under perturbation of the transition matrix is investigated, and examples exhibiting some counter-intuitive behaviour are presented. Finally, we characterize the situation in which the accessibility indices for all states coincide.     

How small are the increments of partial sums of a discrete random field?

In this paper we establish inferior limit results and path properties for the increments of partial sums of a strictly stationary and linearly positive quadrant dependent (LPQD) or linearly negative quadrant dependent (LNQD) discrete random field with multidimensional indices.     

Weak convergence of random sums and maximum random sums under nonrandom norming

Necessary and sufficient conditions are obtained for the weak convergence of random sums and maximum random sums of independent identically distributed random variables. Limit distributions for these sums are described. The indices are not assumed to be independent of the summands. Supported in part by Grants NFW000 and NFW300 from the International Science Foundation and the Russian Government and by Grant No. 93-011-1446 from the Russian Foundation for Fundamental Research. Proceedings of the XVII Seminar on Stability Problems for Stochastic Models, Kazan, Russia, 1995, Part I.     

The fuzzy hyperbolic inequality index associated with fuzzy random variables

The aim of this paper is focussed on the quantification of the extent of the inequality associated with fuzzy-valued random variables in general populations. For this purpose, the fuzzy hyperbolic inequality index associated with general fuzzy random variables is presented and a detailed discussion of some of the most valuable properties of this index (extending those for classical inequality indices) is given. Two examples illustrating the computation of the fuzzy inequality index are also considered. Some comments and suggestions are finally included.     

Convergence of series of probabilities of large deviations for sums of random variables with multidimensional indices

One finds necessary and sufficient conditions for the convergence of series of weighted probabilities of large deviations for sums of independent random variables with multidimensional indices. These conditions are expressed in terms of the initial distribution.Translated from Veroyatnostnye Raspredeleniya i Matematicheskaya Statistika, pp. 114–131, 1986.     

Stability of Maxima of Random Variables with Multidimensional Indices

The paper discusses the stability of suitably-defined maxima of a set of i.i.d. random variables with multidimensional indices.It is shown that theorems of Gnedenko (1943) and Tomkins (1986) concerning relative stability and complete relative stability of maxima remain valid in the new setting.Moreover, a criterion for almost sure relative stability for maxima with multidimensional indices is presented, extending a result of Barndorff-Nielsen (1963).AMS 2000 Subject Classification. Primary—60F15, 60G60, 62G30, Secondary—10A25, 60G99     

Multi-armed bandits with simple arms

An exact solution to certain multi-armed bandit problems with independent and simple arms is presented. An arm is simple if the observations associated with the arm have one of two distributions conditional on the value of an unknown dichotomous parameter. This solution is obtained relating Gittins indices for the arms to ladder variables for associated random walks.     

Lil and SLLN for random sums

In this paper, we prove an inverse of the law of the iterated logarithm for a randomly indexed sequence of random variables with indices assumed to be independent of the basic sequence. We also prove that the Borel-Cantelli version of the Anscombe condition, introduced by A. Gut, is necessary and sufficient for the complete convergence (in the Hsu-Robbins sense) of a randomly indexed sequence of random variables. Written during the Ph.D. study at the Department of Computational Mathematics and Cybernetics, Moscow State University. Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part I, Eger, Hungary, 1994.