Randomly weighted series of iid's in $L^1$

Abstract

Let $(Y_k)$ be an integrable sequence of iid random variables defined on the probability space $(Y,\mathcal F, \mu)$. We prove that there exists a subset $Y^{*}\subset Y$ of full measure such that for each $y\in Y^{*}$ the following holds: for every integrable iid sequence $(X_k)$ on a probability space $(X,\Sigma,m)$, the series

$$\lim_{n\to\infty}\sideset{}{'}\sum_{k=-n}^{n}\frac{Y_k(y)X_k(x)}{k}$$

converges for almost every $x\in X$.