一类统计量的乘积的渐近性质和几乎处处中心极限定理

The Asymptotic Properties and Almost Sure Central Limit Theorems for the Products of a Class of Statistics

设{X_n,-∞<n<∞}为独立同分布平方可积正值随机变量序列,u=EX_1,σ~2=VarX_1>0.记S_n=sum from X_i,T_n=T-n(X_1,…,X_n)是一统计量(或随机函数),可被表示为T_n=a_nS_n+R_n,其中a_n>0为常数序列,R_n为余项.该文证明若R_n=o(a_nn^(1/2))a.s.,则对统计量T_n的乘积的几乎处处中心极限定理成立,且给出了它的渐近分布和弱不变原理.并以U统计量,Von-Mises统计量,线性模型误差方差的估计等几个常见的统计量为例说明结果应用的广泛性.推广了以往文献中关于独立同分布随机变量和的乘积及U统计量乘积的相应结果.

Let {Xn, -∞〈 n 〈 ∞} be a sequence of independent and identically distributed, positive, square integrable random variables with # = EX1, cr2 = VarX1 〉 O. The asymptotic properties for the products of a class of statistics （or random functions） expressed by Tn = anS, ＋ Rn are discussed, where Sn = ∑i=1^n Xi, an 〉 0 is a sequence of constants, Rn = o（an√n） i=1 a.s.. The results contain the almost sure central limit theorems, asymptotically lognormality and the weak invariance principles. Some examples such as U-statistics, Von-Mises statistics, error variance estimates in linear models are stated to illustrate the generality of the results.