关于修整和强逼近的一个注记

A Note on Strong Approximation for Trimmed Sums

设{X,X_n;n≥1}是一独立同分布的随机变量序列.如果|X_m|是新序列{|X_k|;k≤n}中的第r大元素,则令X_n^((r)=X_m.同时记部分和与修整和分别为S_n=sum from k=1 to n X_k和^((r))S_n=S_n-(X_n^((1))+…+X_n^((r))).该文在EX^2可能是无穷的条件下,得到了修整和^((r))S_n的广义强逼近定理.作为应用,建立了关于修整和以及修整和乘积的广义泛函重对数律.

Let {X, Xn; n ≥ 1} be a sequence of independent and identically distributed random variables, and let X（r） = Xm if ｜Xm｜ is the r-th maximum of {｜Xk｜; k ≤ n}. Define Sn=∑k=1^nXk and （r）Sn=Sn-（Xn^（1）＋…＋Xn^（r）=Xm.. This paper aims to establish a general strong approximation for the trimmed sums （r）Sn without variance, and as applications, general functional laws of the iterated logarithm for trimmed sums and products of trimmed sums are derived.