Almost sure local limit theorems

C HUNG E RDÖS (1951) are among the first to prove some form of an almost sure local limit theorem (cf. C SÁKI et al., 1993). Here we propose a formulation of such statements and discuss related problems.     

On random fuzzy variables of second order and their application to linear statistical inference with fuzzy data

This paper summarizes some results on random fuzzy variables with existing expectation and variance, called random fuzzy variables of second order. Using the Frechét-principle and – via support functions – the embedding of convex fuzzy sets into a Banach space of functions it especially presents a unified view on expectation and variance of random fuzzy variables. These notions are applied in developing linear statistical inference with fuzzy data. Detailed investigations are presented concerning best linear unbiased estimation in linear regression models with fuzzy observations. Received: November 1999     

Strong laws for weighted sums of NA random variables

Strong laws are established for linear statistics that are weighted sums of an negatively associated (NA) random sample. The results obtained not only generalize the results of Sung (Stat. Probab. Lett. 52:413–419, 2001) to NA random variables, but also extend and sharpen them. Research supported by Youth Talent Foundation of Zhejiang Gongshang University (Q07-07).     

Sociological Models Based on Fréchet Random Variables

Products of random variables are of both practical and theoretical significance to social scientists. This has increased the need to have available the widest possible range of statistical results on products of random variables. In this note, the distribution of the product XY is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.     

A note on mixed Poisson approximation

Conditions sufficient for the weak convergence of sums of 0–1 random variables indexed by k -sets to a mixed Poisson law are presented. An example of applications concerning "distance" random variables is also provided.     

无穷级数求和方法研究

任何一个级数要么收敛要么发散,收敛的级数都是应该有和或和函数的。无穷级数求和方法很多,有很强的技巧性,文章就通过例子介绍无穷级数求和的若干方法,如裂项相消法、逐项微分或积分法、转化为函数项级数求解法、利用子列的极限等等,其目的是让学习者更加熟练地掌握无穷级数求和方法及技巧,从而进一步促进其对该知识的学习和理解。     

On the Ratio of Fréchet Random Variables

The distribution of the ratio X/Y is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.     

Slow convergence of the number of near-maxima for Burr XII distributions

A near-maximum is an observation which falls within a distance a of the maximum observation in an independent and identically distributed sample of size n. Subject to some conditions on the tail thickness of the population distribution, the number K _n (a) of near-maxima is known to converge in probability to one or infinity, or in distribution to a shifted geometric law. In this paper we show that for all Burr XII distributions K _n (a) converges almost surely to unity, but this convergence property may not become clear under certain cases even for very large n. We explore the reason of such slow convergence by studying a distributional continuity between Burr XII and Weibull distributions. We have also given a theoretical explanation of slow convergence of K _n (a) for the Burr XII distributions by showing that the rate of convergence in terms of P{K _n (a) > 1} tending to zero changes very little with the sample size n. Illustrations of the limiting behaviour K _n (a) for the Burr XII and the Weibull distributions are given by simulations and real data. The study also raises an important issue that although the Burr XII provides overall better fit to a given data set than the Weibull distribution, cautions should be taken for the extrapolation of the upper tail behaviour in the case of slow convergence.      

正项级数收敛性的判别法的进一步研究

用初等方法深入研究了正向级数判别法,基于比值判别法和根值判别法判别法思想,研究了更高精度判别法,给出不断提高精度的判别法的构造思想以及一般性定理。     

The First Two Eigenvalues of Large Random Matrices and Brody's Hypothesis on the Stability of Large Input–Output Systems

Bródy (1997) notices that for large random Leontief matrices, namely non-negative square matrices with all entries i.i.d., the ratio between the subdominant eigenvalue (in modulus) and the dominant eigenvalue declines generically to zero at a speed of the square root of the size of the matrix as the matrix size goes to infinity. Since then, several studies have been published in this journal in attempting to rigorously verify Bródy's conjecture. This short article, drawing upon some theorems obtained in recent years in the literature on empirical spectral distribution of random matrices, offers a short proof of Bródy's conjecture, and discusses briefly some related issues.     

The ascending ladder height distribution for a certain class of dependent random walks

A random walk { S_n } with S_n = (X_l - Y_l) +…+ ( X_n - Y_n ) is considered where the X_n Y_n are non-negative random variables, the Y_n are exponentially distributed with rate δ and the X_n have common distribution function B . It is shown that the expression δ(1 - S (x)) for the density of the ascending ladder height distribution of (S_n), which is well-known for i.i.d. X_n , holds also when the X_n form a stationary sequence of not necessarily independent random variables.     

Stability and self-decomposability of semi-group valued random variables

A random variable X on IR₊ is said to be self-decomposable, dif for all c∈ (0, 1) there exists a random variable X_c on IR₊ such that X=^dcX+X_c . It is said to be stable if it is self-decomposable and X_c=^d (1 - c)X' , where X and X' are identically and independently distributed. The notions of stability and self-decomposability for infinitely divisible random variables are generalised to abelian semi-groups ( S, + ) with S having an identical involution, by using characteristic functions. The generalised definitions involve semi-groups of scaling operators T . There operators can be interpreted in a slightly different context as generalised continuous-time branching processes (with immigration). The underlying importance of the generator of the semi-groups T in the characterisation of stability and self-decomposability is stressed.     

On some properties of the linear function Poisson binomial and negative binomial distributions

Summary A general model in fluctuations of sums of random variables leading, under certain assumptions, to each of the generalized and linear function Poisson, binomial and negative binomial distributions is presented. Moreover the generating functions and the factorial moments of the linear function Poisson, binomial and negative binomial distributions are obtained in close forms and certain distributional properties are discussed.