Monotonicity and Asymptotic Queue-Length Distribution in Discrete-Time Networks

For closed, cyclic, discrete-time networks with one server per node and with independent, geometric service times, in equilibrium, the joint queue-length distribution can be realized as the joint distribution of independent random variables, conditionally given their sum. This tool helps establish monotonicity properties of performance measures and also helps show that the queue-length random variables are negatively associated. The queue length at a node is asymptotically analyzed through a family of networks with a fixed number of node types, where the number of nodes approaches infinity, the ratio of jobs to nodes has a positive limit, and each node type has a limiting density. The queue-length distribution at any node is shown to converge, in a strong sense, to a distribution that is conditionally geometric. As a by-product, this approach settles open issues regarding occupancy proportion and average queue length at a node type.     

Some Asymptotic Results for One-Sided Large Deviation Probabilities

We investigate the asymptotic behavior of the sum of independent real random variables. We assume that the random variables are not identically distributed but the average of distribution functions of these random variables is equivalent to some heavy-tailed limit distribution function. An example with Pareto law as limit function is given.     

On the distribution of the (un)bounded sum of random variables

We propose a general treatment of random variables aggregation accounting for the dependence among variables and bounded or unbounded support of their sum. The approach is based on the extension to the concept of convolution to dependent variables, involving copula functions. We show that some classes of copula functions (such as Marshall-Olkin and elliptical) cannot be used to represent the dependence structure of two variables whose sum is bounded, while Archimedean copulas can be applied only if the generator becomes linear beyond some point. As for the application, we study the problem of capital allocation between risks when the sum of losses is bounded.     

A class of models for uncorrelated random variables

We consider the class of multivariate distributions that gives the distribution of the sum of uncorrelated random variables by the product of their marginal distributions. This class is defined by a representation of the assumption of sub-independence, formulated previously in terms of the characteristic function and convolution, as a weaker assumption than independence for derivation of the distribution of the sum of random variables. The new representation is in terms of stochastic equivalence and the class of distributions is referred to as the summable uncorrelated marginals (SUM) distributions. The SUM distributions can be used as models for the joint distribution of uncorrelated random variables, irrespective of the strength of dependence between them. We provide a method for the construction of bivariate SUM distributions through linking any pair of identical symmetric probability density functions. We also give a formula for measuring the strength of dependence of the SUM models. A final result shows that under the condition of positive or negative orthant dependence, the SUM property implies independence.     

Asymptotic Results for the Sum of Dependent Non-identically Distributed Random Variables

In this paper we extend some results about the probability that the sum of n dependent subexponential random variables exceeds a given threshold u. In particular, the case of non-identically distributed and not necessarily positive random variables is investigated. Furthermore we establish criteria how far the tail of the marginal distribution of an individual summand may deviate from the others so that it still influences the asymptotic behavior of the sum. Finally we explicitly construct a dependence structure for which, even for regularly varying marginal distributions, no asymptotic limit of the tail of the sum exists. Some explicit calculations for diagonal copulas and t-copulas are given. Dominik Kortschak was supported by the Austrian Science Fund Project P18392.     

Characterizing a comonotonic random vector by the distribution of the sum of its components

In this article, we characterize comonotonicity and related dependence structures among several random variables by the distribution of their sum. First we prove that if the sum has the same distribution as the corresponding comonotonic sum, then the underlying random variables must be comonotonic as long as each of them is integrable. In the literature, this result is only known to be true if either each random variable is square integrable or possesses a continuous distribution function. We then study the situation when the distribution of the sum only coincides with the corresponding comonotonic sum in the tail. This leads to the dependence structure known as tail comonotonicity. Finally, by establishing some new results concerning convex order, we show that comonotonicity can also be characterized by expected utility and distortion risk measures.     

相互独立的m值随机变量和的极限分布定理

利用ｍ值随机变量的特征函数,在一定条件下,得到了相互独立的ｍ值随机变量和的极限分布均匀的充要条件;再结合无穷乘积的有关性质,给出了相互独立的ｍ值随机变量和极限分布均匀分布的充分条件,特别当ｍ为素数Ｐ时,所得的充分条件易于验证,且不难满足。     

A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.     

Coupling and Poisson approximation

We give an overview of the Stein-Chen method for establishing Poisson approximations of various random variables. Couplings of certain variables are used to gives explicit bounds for the total variation distance between the distribution of a random variable and a Poisson variable. Some applications are given. In some cases, explicit couplings may be used to obtain good estimates; in other applications it suffices to show the existence of couplings with certain monotonicity properties.Supported by the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine.     

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.     

On the remained term for the central limit theorem

Summary An upper bound for the remainder term of the Edgeworth expansion for the distribution of the normalized sum of independent and identically distributed random variables is given in terms of 3rd and 4th order moments, together with the total variation of the probability density function of the underlying distribution. The Institute of Statistical mathematics     

Series representations for densities functions of a family of distributions—Application to sums of independent random variables

Series representations for several density functions are obtained as mixtures of generalized gamma distributions with discrete mass probability weights, by using the exponential expansion and the binomial theorem. Based on these results, approximations based on mixtures of generalized gamma distributions are proposed to approximate the distribution of the sum of independent random variables, which may not be identically distributed. The applicability of the proposed approximations are illustrated for the sum of independent Rayleigh random variables, the sum of independent gamma random variables, and the sum of independent Weibull random variables. Numerical studies are presented to assess the precision of these approximations.     

A central limit theorem for fuzzy random variables

Fuzzy random variables have been proposed to treat situations in which both random behavior and fuzzy perception must be considered. A definition of independence is given for fuzzy random variables, as well as a notion of fuzzy Gaussian random variables. It is shown that a sum or mean of independent fuzzy random variables converges in the limit to a fuzzy Gaussian random variable, thus providing a fuzzy analogue of the central limit theorem of classical probability theory.     

An alternative proof for the derivation of the joint distribution of a set of linear functions of correlated normal random variables

A single proof is given for the joint distribution of a set of linear functions of correlated normal random variables. The proof avoids using the characteristic function and does not require the transformation matrix to be non-singular.     

独立的无界随机变量和的概率不等式

研究了在概率空间(Ω,T,P)上,独立的无界随机变量和尾部概率不等式,提出了一种用切割原始概率空间(Ω,T,P)的新型方法去处理独立的无界随机变量和。给出了独立的无界随机变量和的指数型概率不等式。作为结果的应用,一些有趣的例子被给出。这些例子表明:文中提出的方法和结果对研究独立的无界随机变量和的大样本性质是十分有用的。     

有趣的随机变量和的分布算法

根据实际问题.提出计算随机变量和的分布的两种方法,即多项式相乘法和概率母函数法.     

Analysis of a three node queueing network

The three node Jackson queueing network is the simplest acyclic network in which in equilibrium the sojourn times of a customer at each of the nodes are dependent. We show that assuming the individual sojourn times are independent provides a good approximation to the total sojourn time. This is done by simulating the network and showing that the sojourn times generally pass a Kolmogorov-Smirnov test as having come from the approximating distribution. Since the sum of dependent random variables may have the same distribution as the sum of independent random variables with the same marginal distributions, it is conceivable that our approximation is exact. However, we numerically compute upper and lower bounds for the distribution of the total sojourn time; these bounds are so close that the approximating distribution lies outside of the bounds. Thus, the bounds are accurate enough to distinguish between the two distributions even though the Kolmogorov-Smirnov test generally cannot.     

On Edgeworth Expansions in Generalized Urn Models

The random vector of frequencies in a generalized urn model can be viewed as conditionally independent random variables, given their sum. Such a representation is exploited here to derive Edgeworth expansions for a “sum of functions of such frequencies,” which are also called “decomposable statistics.” Applying these results to urn models such as with- and without-replacement sampling schemes as well as the multicolor Pólya–Egenberger model, new results are obtained for the chi-square statistic, for the sample sum in a without-replacement scheme, and for the so-called Dixon statistic that is useful in comparing two samples.     

On the convolution of Mittag–Leffler distributions and its applications to fractional point processes

We obtain the distribution of the sum of independent Mittag–Leffler (ML) random variables which are not necessarily identically distributed. Firstly we discuss the corresponding known result for independent and identically distributed ML random variables which follows as a special case of our result. Some applications of the obtained result to fractional point processes are also discussed.     

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.