k-U统计量的指数收敛速度

研究了k-U统计量的收敛速度,在一组适当的正则条件下,获得了k-U统计量的指数收敛速度,推广了U-统计量的指数收敛速度的相应结果.     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

任意随机变量序列级数的强收敛性

利用鞅差序列级数的收敛定理和条件三级数定理研究了任意随机变量序列级数的强收敛性,推广了某些经典的鞅差序列和独立随机变量序列及两两NQD序列的强极限定理.     

Rates of convergence for the laws of large numbers for independent Banach-valued random variables

In this paper, results of Lai, Heyde, and Rohatgi concerning the convergence rates for the laws of large numbers are extended for the case of independent random variables taking values in a separable Banach space.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

Rates of convergence of quadratic rank statistics

The rates of convergence of the distribution function of quadratic rank statistics to the X²-distribution under hypothesis and near alternatives are investigated. The considered quadratic rank statistics are used for testing the multivariate hypothesis of randomness. The method suggested by Jure?ková [7] is applied.     

On the complete convergence of sums of negatively associated random variables

This paper extends results on complete convergence in the law of large numbers for subsequences to the case of negatively associated nonidentically distributed random variables. Translated fromMatematicheskie Zametki, Vol. 68, No. 3, pp. 411–420, September, 2000.     

ALMOST SURE CONVERGENCE OF THE STABLE TAIL EMPIRICAL DEPENDENCE FUNCTION IN MULTIVARIATE EXTREME STATISTICS

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionandTheoremsSupposethatF(x,y)isabivariatedistributionfunctionwithtwocontinuousmarginaldistributionfunctions,say,FIandF2.DefineFissaidtohaveastabletaildependencefunction(STDF)l(x,y)ifforx20andy20,whereF(x,y)~1--F(QI(x),QZ(y)).TheconceptofSTDFwasintroducedin[6].Supposethat{(Xi,K),i21}isasequenceofi.i.d.randomvectorswithdistributionF(x,y).Ifthereedestsomesequencesofconstantsan>0,on>0,b.ERandd.ER,n>1.suc…     

An extension of almost sure central limit theorem for order statistics

Let {X _n ,n ≥ 1} be a sequence of i.i.d. random variables. Let M _n and m _n denote the first and the second largest maxima. Assume that there are normalizing sequences a _n  > 0, b _n and a nondegenerate limit distribution G, such that . Assume also that {d _k ,k ≥ 1} are positive weights obeying some mild conditions. Then for x > y we have

不同分布(φ)混合随机变量序列的强收敛性

本文研究了不同分布(φ)混合随机变量序列的强收敛性质的问题.利用(φ)混合随机变量序列的矩不等式和截尾的方法,获得了(φ)混合随机变量序列完全收敛性和几乎处处收敛性结果,所获得结果不仅推广了Baum和Katz (1965)关于独立同分布随机变量序列的结论,而且改进了Wu和Lin (2004)关于同分布(φ)混合随机变量序列的相关结论.     

The uniform convergence and precise asymptotics of generalized stochastic order statistics

Let X(i,n,m,k), i=1,…,n, be generalized order statistics based on F. For fixed r∈N, and a suitable counting process N(t), t>0, we mainly discuss the precise asymptotic of the generalized stochastic order statistics X(N(n)−r+1,N(n),m,k). It not only makes the results of Yan, Wang and Cheng [J.G. Yan, Y.B. Wang, F.Y. Cheng, Precise asymptotics for order statistics of a non-random sample and a random sample, J. Systems Sci. Math. Sci. 26 (2) (2006) 237-244] as the special case of our result, and presents many groups of weighted functions and boundary functions, but also permits a unified approach to several models of ordered random variables.     

Limit distributions of upper order statistics for families of multivariate distributions

We consider a portfolio of dependent exchangeable random variables , where the dependence structure is generated by a mixture model (Archimedean copulas belong to this class of models). Define the ordered sample . We prove results of the following type: fix and choose appropriately, then converges in distribution to a random vector as , for which we can explicitly give the distribution.     

A Supplement to the Baum-Katz-Spitzer Complete Convergence Theorem

Let {X, Xn; n≥ 1} be a sequence of i.i.d. Banach space valued random variables and let {an; n ≥ 1} be a sequence of positive constants such that an↑∞ and 1〈 lim inf n→∞ a2n/an≤lim sup n→∞ a2n/an〈∞ Set Sn=∑i=1^n Xi,n≥1.In this paper we prove that ∑n≥1 1/n P（｜｜Sn｜｜≥εan）〈∞ for all ε〉0 if and only if lim n→∞ Sn/an=0 a.s. This result generalizes the Baum-Katz-Spitzer complete convergence theorem. Combining our result and a corollary of Einmahl and Li, we solve a conjecture posed by Gut.     

Random series in using unconditional basic sequences and stable sequences: A result on almost sure almost everywhere convergence

Here we study the almost sure almost everywhere convergence of random series of the form in the Lebesgue spaces , where the 's are centered random variables, and the 's constitute an unconditional basic sequence or an stable sequence. We show that if one of these series converges in the norm topology almost surely, then it converges almost everywhere almost surely.

     

Almost sure convergence theorem and strong stability for weighted sums of NSD random variables

In this paper, Kolmogorov-type inequality for negatively superadditive dependent (NSD) random variables is established. By using this inequality, we obtain the almost sure convergence for NSD sequences, which extends the corresponding results for independent sequences and negatively associated (NA) sequences. In addition, the strong stability for weighted sums of NSD random variables is studied.     

可交换随机变量序列重对数律的收敛速度

本文讨论了可交换随机变量序列{Xn:n≥1)重对数律的收敛速度,得到了可交换随机变量序列与独立序列类似的极限性质,同时给出了可交换序列重对数律收敛速度的一种描述.     

On the rate of convergence to normality for generalized linear rank statistics

Summary A generalized linear rank statistic is introduced to include, as special cases, both signed as well as unsigned linear rank statistics. For this statistic, the rate of convergence to asymptotic normality is investigated. It is shown that this rate is of orderO(N ^−1/2 logN) if the score generating function ϕ is twice differentiable, and it is of orderO(N ^−1/2) if the second derivative of ϕ satisfies Lipschitz's condition of order ≧1/2. The results obtained extend as well as generalize most of the earlier results obtained in this direction.     

Criteria for Convergence of the Number of Near Maxima for Long Tails

Let M_n denote the maximum of a random sample of size n and K_n(a) be the number of near maxima, i.e. the number of sample observations in the fixed-width window (M_n – a, M_n]. There is a known integral criterion for almost sure convergence (to unity) of K_n(a), and we establish a similar criterion for complete convergence. We obtain simple but quite general sufficient conditions on the survivor function for satisfying the integral criteria. Further insight is obtained by seeking the rate at which P(K_n(a > 1)) tends to zero.AMS 2000 Subject Classification. 62G30, 60F15     

Rates of uniform convergence of extreme order statistics

Summary Bounds for the convergence uniformly over all Borel sets of the largest order statistic as well as of the joint distribution of extremes are established which reveal in which way these rates are determined by the distance of the underlying density from the density of the corresponding generalized Pareto distribution. The results are highlighted by several examples among which there is a bound for the rate at which the joint distribution of thek largest order statistics from a normal distribution converges uniformly to its limit.