Assume that $X_n^N$, $n=1,\ldots,N$ for all $N>0$ are 0-1 random variables such that $$ \frac{1}{N} \sum_{n=1}^N X_n^N \to p $$ almost surely. Now, consider $a$ sequence of i.i.d. Bernoulli random variables $(Y_n)$ with parameter $q$. It is known that $Y_n$ is independent of $X_n$, and that $X_n$ may depend on $Y_i$ but only if $i<n$.

My question is: can we conclude that $$ \frac{1}{N} \sum_{n=1}^N X_n^N Y_n \to pq $$ almost surely?

My attempt/note: if the LHS would be replaced by expectation, then the result would be true because $\mathbb{E}[X_n Y_n] = \mathbb{E}[X_n] q$ and $\frac{1}{N} \sum_{n=1}^N \mathbb{E}[X_n] \to p$ by the uniformly integrability of the $X_n^N$. This should also mean that if the random variable $\frac{1}{N} \sum_{n=1}^N X_n^N Y_n$ converges, then it must converge to $pq$. So it would remain to show that it converges.


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