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Let    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\{X_{n}, n\geq1\}$\end{document}   {   X  n   ,  n  ≥  1  }be a strictly stationary negatively associated sequence of positive random variables with    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\mathrm{E}X_{1}=\mu>0$\end{document}   E   X  1   =  μ  >  0and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\operatorname{Var}(X_{1})=\sigma^{2}<\infty$\end{document}   Var  (   X  1   )  =   σ  2   <  ∞. Denote    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$S_{n}=\sum_{i=1}^{n}X_{i}, p_{k}=\mathrm{P}(a_{k}\leq ({\prod}_{j=1}^{k}S_{j}/(k!\mu^{k}) )^{1/(\gamma\sigma_{1} \sqrt{k})}< b_{k})$\end{document}    S  n   =   ∑   i  =  1   n    X  i   ,   p  k   =  P  (   a  k   ≤    (   ∏   j  =  1   k    S  j   /  (  k  !   μ  k   )  )    1  /  (  γ   σ  1    k   )    <   b  k   )and    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\gamma=\sigma/\mu$\end{document}   γ  =  σ  /  μthe coefficient of variation. Under some suitable conditions, we derive the almost sure local central limit theorem    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$$\lim_{n\rightarrow\infty}\frac{1}{\log n}\sum_{k=1}^{n} \frac{1}{kp_{k}}\mathrm{I} \biggl\{ a_{k}\leq \biggl(\frac {\prod_{j=1}^{k}S_{j}}{k!\mu^{k}} \biggr)^{1/(\gamma\sigma_{1} \sqrt {k})}< b_{k} \biggr\} =1 \quad\mbox{a.s.,} $$\end{document}    lim   n  →  ∞1   log  n∑   k  =  1   n    1   k   p  kI   {   a  k   ≤    (∏   j  =  1   k    S  jk  !   μ  k)    1  /  (  γ   σ  1    k   )    <   b  k   }   =  1    a.s.,where    \documentclass[12pt]{minimal} 				\usepackage{amsmath} 				\usepackage{wasysym}  				\usepackage{amsfonts}  				\usepackage{amssymb}  				\usepackage{amsbsy} 				\usepackage{mathrsfs} 				\usepackage{upgreek} 				\setlength{\oddsidemargin}{-69pt} 				\begin{document}$\sigma_{1}^{2}=1+\frac{1}{\sigma^{2}}\sum_{j=2}^{\infty}\operatorname{Cov}(X_{1},X_{j})>0$\end{document}    σ   1   2   =  1  +   1   σ  2∑   j  =  2   ∞   Cov  (   X  1   ,   X  j   )  >  0.

The almost sure local central limit theorem for products of partial sums under negative association