Consider a sequence of i.i.d. random variables $\left\{X_i\right\}_i$, and let $Y$ be another random variable.
Can we say something regard the convergence of the following series $$ \frac{1}{n}\sum_{i=1}^nf\left(X_iY\right) $$ as $n\to\infty$ (assume that $f$ is some "nice" function) ? If $Y$ was not there, then we obviously could use the law of large numbers to claim that the above sum converge in probability to $E(f(X))$. But, with $Y$, the summands are dependent and thus we cannot directly apply the law of large numbers, right?
EDIT: The answer to this question is as follows (credit to Did):
If one assumes that $Y$ is independent on the sequence $(X_i)$, then it can be shown that the series converges a.s. to the conditional expectation $E(f(X_1Y)\mid Y)$. This can be shown by using backwards martingale convergence theorem (actually a refined proof of the SLLN using the backwards martingale convergence theorem).
If, however, $Y$ do depend on the the sequence $(X_i)$, then there is nothing much to say (in general).