Communications of the
Korean Mathematical Society CKMS

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