The limit theorems under logarithmic averages for mixing random variables
Commun. Korean Math. Soc. 2014 Vol. 29, No. 2, 351-358
Printed April 1, 2014
Yong Zhang
Jilin University
Abstract : In this paper, under some suitable integrability and smoothness conditions on $f$, we establish the central limit theorems for $$\sum_{k\leq N}k^{-1} f(S_k/{\sigma\sqrt{k}}),$$ where $S_k$ is the partial sums of strictly stationary mixing random variables with $EX_1=0$ and $\sigma^2=EX_1^2+2\sum_{k=1}^{\infty}EX_1X_{1+k}$. We also establish an almost sure limit behaviors of the above sums.
Keywords : central limit theorem, almost sure central limit theorem, logarithmic averages, mixing random variables
MSC numbers : Primary 60F15; Secondary 60F05
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