I have the following sequence $\{Y_n\}$ of random variables defined as $$Y_n=\frac{\sum_{k=1}^{n}a^{k-1}X_k}{\sum_{k=1}^{n}a^{k-1}}$$ where $\{X_n\}$ is a sequence of i.i.d. random variables.
My question is, is there any weak or strong law of large numbers that says what $Y_n$ is going to converge to, if at all it converges?
I can understand that in general the statement $Y_n\stackrel{p}{\to}\mathbb{E}X$ is incorrect. An illustration of this fact can be given by taking $X_n$'s as i.i.d standard normal random variables, which shows that $Y_n\stackrel{a.s.}{\to}Y$ where $Y\sim \mathcal{N}\left(0,\frac{1-a}{1+a}\right)$ with the assumption that $|a|<1$. But I am unable to get the limit in general. Please help. Thanks in advance.
I'm going to assume that $\Bbb E|X_1|<\infty$.
1. If $|a|<1$ then the random series $\sum_{k=1}^\infty a^{k-1}X_k$ converges in $L^1$, hence in probability, hence a.s. (by a theorem of P. Lévy). Consequently $Y_n$ converges a.s to $(1-a)\sum_{k=1}^\infty a^{k-1}X_k$.
2. If $|a|>1$ then $Y_n=(1-a^{-1})\sum_{k=1}^n (a^{-1})^{k-1}X_{n-k+1}$, which has the same distribution as $(1-a^{-1})\sum_{k=1}^n (a^{-1})^{k-1}X_{k}$, which converges in distribution to $(1-a^{-1})\sum_{k=1}^\infty (a^{-1})^{k-1}X_{k}$, by the analysis for 1.