Let {Xn,-∞＜ n ＜∞} be a sequence of independent identically distributed be a random function such that Tn = ASn+ Rn,where supnE|Rn|＜∞ and Rn = o(√n)a.s.,or Rn = O(n1/2-2γ) a.s.,0 ＜γ＜ 1/8.In this paper,we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn.As a consequence,it can be shown that ASCLT and FASCLT also hold for U-statistics,Von-Mises statistics,linear processes,moving average processes,error variance estimates in linear models,power sums,product-limit estimators of a continuous distribution,product-limit estimators of a quantile function,etc.
LU Chuanrong, QIU Jin & XU Jianjun School of Mathematics and Statistics, Zhejiang University of Finance and Economics, Hangzhou 310018, China
