In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather ar...In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences. This extends the earlier work on mixing random variables展开更多
Let{Xn,-∞<n<∞}be a sequence of independent identically distributed random variables with EX1=0,EX12=1 and let Sn=∑k=1∞Xk,and Tn=Tn(X1,…,Xn)be a random function such that Tn=ASn+Rn,where supn E|Rn|<∞and ...Let{Xn,-∞<n<∞}be a sequence of independent identically distributed random variables with EX1=0,EX12=1 and let Sn=∑k=1∞Xk,and Tn=Tn(X1,…,Xn)be a random function such that Tn=ASn+Rn,where supn E|Rn|<∞and Rn=o(n^(1/2))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.展开更多
文摘In this paper, we prove an almost sure central limit theorem for weighted sums of mixing sequences of random variables without stationary assumptions. We no longer restrict to logarithmic averages, but allow rather arbitrary weight sequences. This extends the earlier work on mixing random variables
基金This work was partially supported by the Natural Science Foundation of Zhejiang Province(Grant No.101016)the National Natural Science Foundation of China(Grant No.10471126).
文摘Let{Xn,-∞<n<∞}be a sequence of independent identically distributed random variables with EX1=0,EX12=1 and let Sn=∑k=1∞Xk,and Tn=Tn(X1,…,Xn)be a random function such that Tn=ASn+Rn,where supn E|Rn|<∞and Rn=o(n^(1/2))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.
