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- Author or Editor: Khurelbaatar Gonchigdanzan x

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Summary In this note we prove an almost sure limit theorem for the products of *U*-statistics.

## Abstract

Let {

*X*:_{n}*n*≧ 1} be a sequence of dependent random variables and let {*w*: 1 ≦_{nk}*k*≦*n, n*≧ 1} be a triangular array of real numbers. We prove the almost sure version of the CLT proved by Peligrad and Utev [7] for weighted partial sums of mixing and associated sequences of random variables, i.e.$$\underset{n\to \infty}{\mathrm{lim}}\frac{1}{\mathrm{log}\text{\hspace{0.17em}}n}{\displaystyle \sum _{k=1}^{n}\frac{1}{k}\text{I}\left({\displaystyle \sum _{i=1}^{k}{w}_{ki}{X}_{i}\leqq}x\right)=\frac{1}{\sqrt{2\pi}}{\displaystyle \underset{-\infty}{\overset{x}{\int}}{e}^{-\frac{1}{2}{t}^{2}}dt\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{a}\text{.s}\mathrm{..}}}$$