SCHWEITZER 1979 Problem 11

Let $\{\xi_{k \ell} \}_{k,\ell=1}^{\infty}$ be a double sequence of random variables such that

$$\Bbb{E}( \xi_{ij} \xi_{k\ell})= \mathcal{O} \left(\frac{ \log(2|i-k|+2)}{ \log(2|j-\ell|+2)^{2}}\right) \;\;\;(i,j,k,\ell =1,2, \ldots ) \ .$$

Prove that with probability one,

$$\frac{1}{mn} \sum_{k=1}^m \sum_{\ell=1}^n \xi_{k\ell} \rightarrow 0 \;\;\textrm{as} \max (m,n)\rightarrow \infty\ .$$

F. Moricz