A note on runs of geometrically distributed random variables

Recently, Grabner et al. [Combinatorics of geometrically distributed random variables: run statistics, Theoret. Comput. Sci. 297 (2003) 261-270] and Louchard and Prodinger [Ascending runs of sequences of geometrically distributed random variables: a probabilistic analysis, Theoret. Comput. Sci. 304 (2003) 59-86] considered the run statistics of geometrically distributed independent random variables. They investigated the asymptotic properties of the number of runs and the longest run using the corresponding probability generating functions and a Markov chain approach. In this note, we reconsider the asymptotic properties of such statistics using another approach. Our approach of finding the asymptotic distributions is based on the construction of runs in a sequence of m-dependent random variables. This approach enables us to find the asymptotic distributions of many run statistics via the theorems established for m-dependent sequence of random variables. We also provide the asymptotic distribution of the total number of non-decreasing runs and the longest non-decreasing run.     

The Convergence of the Sums of Independent Random Variables Under the Sub-linear Expectations

Let {Xn;n≥1} be a sequence of independent random variables on a probability space(Ω,F,P) and Sn=∑k=1n Xk.It is well-known that the almost sure convergence,the convergence in probability and the convergence in distribution of Sn are equivalent.In this paper,we prove similar results for the independent random variables under the sub-linear expectations,and give a group of sufficient and necessary conditions for these convergence.For proving the results,the Levy and Kolmogorov maximal inequalities for independent random variables under the sub-linear expectation are established.As an application of the maximal inequalities,the sufficient and necessary conditions for the central limit theorem of independent and identically distributed random variables are also obtained.     

非平稳NA随机变量满足迭对数律及大数定律的充分条件

本文应用Shao所提供的极大值矩不等式及概率不等式,给出了不具有平稳分布的NA随机变量列满足迭对数律和大数定律的充分条件.     

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本文给出了上期望空间中独立随机变量部分和的最大不等式、指数 不等式、Marcinkiewicz-Zygmund不等式. 并且应用指数不等式和Marcinkiewicz-Zygmund不等式 研究了随机变量部分和序列完备收敛的性质.     

NA随机变量序列的最大部分和不等式及有界重对数律

本文给出了NA随机变量序列关于最大部分和的概率不等式及矩不等式,并获得了NA随机变量序列的Teicher型和Egorov型有界重对数律等.     

m-NA随机阵列的完全收敛性的一个注记

本文研究了行m-NA随机阵列的完全收敛性.利用文[8]中结果获得了m-NA列最大部分和的一个概率不等式,并根据该不等式和截尾的方法,探讨了行m-NA随机阵列的完全收敛性,获得了与行NA随机阵列情形类似的结果,简化了文[5]中定理1的证明.     

Sensitivity analysis of a railway station track layout with respect to a given timetable

An important step in the process of designing a railway station track layout is the verification of the robustness of the layout with respect to the timetables it is based on. For this purpose we develop in this paper an algorithm to randomly perturb a given timetable such that the perturbation is feasible and has the same structure as the given timetable. Mathematically, in this paper we study the problem of, given a set of integer variables and a set of binary relations stating minimal and maximal differences between the variables, to generate solutions uniformly at random. The algorithm involves the simulation of a Markov chain whose state space is a particular subset of the set of feasible timetables and whose limiting and equilibrium distribution is the uniform distribution. Whereas this idea seems simple, some technical pitfalls need to be overcome to make it sound.     

关于次序统计量数学期望的两个恒等式的简证及其应用

给出了最大最小次序统计量分布函数和密度函数的恒等式,从而在随机变量独立同分布情形下,给出了关于次序统计量数学期望的两个恒等式的简捷证法.最后将几个恒等式应用于威布尔分布的最大次序统计量的分布及其数字特征的计算.     

Bivariate Markov chain embeddable variables of polynomial type

The primary aim of the present article is to provide a general framework for investigating the joint distribution of run length accumulating/enumerating variables by the aid of a Markov chain embedding technique. To achieve that we introduce first a class of bivariate discrete random variables whose joint distribution can be described by the aid of a Markov chain and develop formulae for their joint probability mass function, generating functions and moments. The results are then exploited for the derivation of the distribution of a bivariate run-related statistic. Finally, some interesting uses of our results in reliability theory and educational psychology are highlighted. Research supported by General Secretary of Research and Technology of Greece under grand PENED 2001.     

On the Distribution of Sums of Independent Exponential Random Variables Via Wilks’ Integral Representation

In this note we consider an alternative approach to compute the distribution of the sum of independent exponential random variables. In particular, by considering the logarithmic relation between exponential and beta distribution functions and by considering the Wilks’ integral representation for the product of independent beta random variables, we provide a closed-form expression for the distribution of the sum of independent exponential random variables. The expression we obtain is simpler than the ones previously obtained in the literature.     

平稳NA随机变量序列中偏差的下界估计

本文在比较一般的条件下得到了平稳NA序列的中偏差下界估计，进而得到平稳NA序列的中偏差原理。     

On sums of two counter-monotonic risks

In risk management, capital requirements are most often based on risk measurements of the aggregation of individual risks treated as random variables. The dependence structure between such random variables has a strong impact on the behavior of the aggregate loss. One finds an extensive literature on the study of the sum of comonotonic risks but less, in comparison, has been done regarding the sum of counter-monotonic risks. A crucial result for comonotonic risks is that the Value-at-risk and the Tail Value-at-risk of their sum correspond respectively to the sum of the Value-at-risk and Tail Value-at-risk of the individual risks. In this paper, our main objective is to derive such simple results for the sum of counter-monotonic risks. To do so, we examine separately different contexts in the class of bivariate strictly continuous distributions for which we obtain closed-form expressions for the Value-at-risk and Tail Value-at-risk of the sum of two counter-monotonic risks. The expressions for the subadditive Tail Value-at risk allow us to quantify the maximal diversification benefit. Also, our findings allow us to analyze the tail of the distribution of the sum of two identically subexponentially distributed counter-monotonic random variables.     

On a method of Pollaczek

In 1952, F. Pollaczek gave a method for finding the distribution of the maximal partial sum of mutually independent and identically distributed random variables. In this paper, Pollaczek's results are extended, and the extended results are applied in theory of queues for finding the distribution of the waiting time for single server queues.     

Maximal inequalities for demimartingales and their applications

In this paper, we establish some maximal inequalities for demimartingales which generalize and improve the results of Christofides. The maximal inequalities for demimartingales are used as key inequalities to establish other results including Doob’s type maximal inequality for demimartingales, strong laws of large numbers and growth rate for demimartingales and associated random variables. At last, we give an equivalent condition of uniform integrability for demisubmartingales.     

两两NQD序列部分和的强大数定律

By using the moment inequality, maximal inequality and the truncated method of random variables, we establish the strong law of large numbers of partial sums for pairwise NQD sequences, which extends the corresponding result of pairwise NQD random variables.     

On the comparison of distribution functions of random variables

We prove a rather general comparison principle for the distribution functions of random variables. As a consequence, we obtain a criterion for the equivalence in distribution in the vector sense of an arbitrary sequence of random variables to the Rademacher system; we study the applications of this principle to special cases.     

NA随机变量列的有界重对数律

通过建立NA随机变量最大部分和的一些概率指数不等式，给出了具有不同分布的NA随机变量列有界重对数律的一些结果，因此推广了由R．Wittmann建立的独立随机变量的相关结果。     

A practical non-parametric copula algorithm for system reliability with correlations

System reliability analysis involving correlated random variables is challenging because the failure probability cannot be uniquely determined under the given probability information. This paper proposes a system reliability evaluation method based on non-parametric copulas. The approximated joint probability distribution satisfying the constraints specified by correlations has the maximal relative entropy with respect to the joint probability distribution of independent random variables. Thus the reliability evaluation is unbiased from the perspective of information theory. The estimation of the non-parametric copula parameters from Pearson linear correlation, Spearman rank correlation, and Kendall rank correlation are provided, respectively. The approximated maximum entropy distribution is then integrated with the first and second order system reliability method. Four examples are adopted to illustrate the accuracy and efficiency of the proposed method. It is found that traditional system reliability method encodes excessive dependence information for correlated random variables and the estimated failure probability can be significantly biased.     

ND序列部分和的大偏差和强收敛性

利用ND随机变量序列的矩不等式、极大值不等式以及随机变量的截尾方法,重点研究了ND随机变量序列部分和的大偏差结果和强收敛性,推广了文献中一些相依随机变量序列的若干相应结果.