An improved result in almost sure local central limit theorem for the partial sums

Abstract

The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let $\{X_k,k\ge 1\}$ be a sequence of independent and identically distributed random variables. Under a fairly general condition, an universal result in almost sure local limit theorem for the partial sums $S_k=\sum \nolimits _{i=1}^kX_i$ is established on the weight $d_k=k^{-1}\exp (\log ^\beta k)$, $0\le \beta <1/2$:

$$\begin{aligned}&\underset{n\rightarrow \infty }{\lim } \frac{1}{D_n} \sum \limits _{k=1}^nd_k\frac{\mathrm{I}(a_k\le S_k<b_k)}{\mathrm{P}(a_k\le S_k <b_k)}=1 \ \ \ \mathrm a.s., \end{aligned}$$

where $D_n=\sum \nolimits _{k=1}^nd_k$, $-\infty \le a_k\le 0 \le b_k \le \infty ,\ \ \ k=1,2,\ldots$. This result extends previous results in the almost sure local central limit theorems from $d_k=1/k$ to $d_k=k^{-1}\exp (\log ^\beta k)$, $0\le \beta <1/2$.