Almost sure local central limit theorem for the product of some partial sums with optimized weight

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 (i.i.d.) positive random variables. Under a fairly general condition an universal result in almost sure local limit theorem for the product of some partial sums ${({\prod }_{i=1}^k{S}_{k,i}/({(k-1)}^k{\mu }^k))}^{1/\left(\gamma \sqrt{k}\right)}$ is established on the weight $d_k=k^{-1}\hspace{0.17em}\text{exp}\hspace{0.17em}({\hspace{0.17em}\text{log}\hspace{0.17em}}^{\beta }k),0\le \beta <1/2,$ where $S_{k,i}={\sum }_{j=1}^k{X}_j-X_i,\mathbb{E}{X}_1=\mu$ and the coefficient of variation $\gamma =\sigma /\mu ,$ where $\text{Var}{X}_1={\sigma }^2.$

Keywords:

• Almost sure local central limit theorem
• product
• partial sum
• weight sequences

MATHEMATICS SUBJECT CLASSIFICATION:

• 60F15

Acknowledgments

The authors are also very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.

Additional information

Funding

This work is jointly supported by The National Social Science Fund of China (16BTJ035) and the Natural Science Fund Project of Guangdong Province, China (2016A030313108).