Let $(u_k)_{k\geq 1}$ a sequence of real numbers such that $\lim\limits_{k\to\infty}u_k=+\infty$.
Let $(X_k)_{k\geq 1}$ a sequence of identically distributed random variables, taking their values in $\{0,1\}$, such that $\mathbb{P}(X_1=1)=p\in (0,1)$. Note that I do not assume the $X_k$'s to be independent. Let's say they form a stationary Makov chain.
Do we have, almost surely, $$\lim\limits_{n\to\infty}\frac{1}{n}\sum_{k=1}^nX_ku_k=+\infty$$ ?
If we had $p=1$, this would just be a Cesaro sum and the result would hold. But here we randomly cancel an infinite number of terms. My intuition is that we have this result provided that $u_k$ goes to $+\infty$ fast enough (at least linearly?). However, I don't know how to prove that, as we cannot directly apply the ergodic theorem for Markov chains.