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

The almost sure local central limit theorem is a general result which contains the almost sure global central limit theorem. Let {Xk,k≥1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{X_k,k\ge 1\}$$\end{document} 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 Sk=∑i=1kXi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$S_k=\sum \nolimits _{i=1}^kX_i$$\end{document} is established on the weight dk=k-1exp(logβk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_k=k^{-1}\exp (\log ^\beta k)$$\end{document}, 0≤β<1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \beta <1/2$$\end{document}: limn→∞1Dn∑k=1ndkI(ak≤Sk<bk)P(ak≤Sk<bk)=1a.s.,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\underset{n\rightarrow \infty }{\lim } \frac{1}{D_n} \sum \limits _{k=1}^nd_k\frac{\mathrm{I}(a_k\le S_k<b_k)}{\mathrm{P}(a_k\le S_k <b_k)}=1 \ \ \ \mathrm a.s., \end{aligned}$$\end{document}where Dn=∑k=1ndk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D_n=\sum \nolimits _{k=1}^nd_k$$\end{document}, -∞≤ak≤0≤bk≤∞,k=1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-\infty \le a_k\le 0 \le b_k \le \infty ,\ \ \ k=1,2,\ldots $$\end{document}. This result extends previous results in the almost sure local central limit theorems from dk=1/k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_k=1/k$$\end{document} to dk=k-1exp(logβk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_k=k^{-1}\exp (\log ^\beta k)$$\end{document}, 0≤β<1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0\le \beta <1/2$$\end{document}.