Suppose I have a random variable $X$ which can take values on the set $\mathcal{X}=\{1,2,\dots,m\}$ and $X$ is drawn according to the given probability mass function $\mathbf{p}=\{p_1,p_2,\dots,p_m\}$. Let $S(X)$ be a 'well-behaved' function. Then I want to calculate the following sum:
\begin{equation} \bar{S_n}:=\frac{1}{n} \sum_{k=1}^n \prod_{i=1}^k S(X_i) \tag{1} \end{equation}
where the $X_i$'s are drawn i.i.d $\sim \mathbf{p}$. Explicitly, the average is:
\begin{equation} \bar{S_n}:=\frac{1}{n} \{S(X_1)+S(X_1)S(X_2)+\cdots+S(X_1)\cdots S(X_n)\} \tag{2} \end{equation}
This can be seen as a sum of correlated random variables.
Question. Is there a version of the Law of Large Numbers which can be applied here in order to write the limit of $\bar{S_n}$ in terms of the p.m.f. $\mathbf{p}$?