Let $p\in (0,1)$ and $X_1, X_2, ...X_n \sim \text{Bern}(p)$ be $n$ i.i.d. Bernoulli random variables, where the probability that $X_i$ is $1$ equals $p$.

Fix $a,b>0$ different from $1$ that satisfy $a^p b^{1-p} = 1$, and define $C_i = X_i(a-b)+b$. In other words, $C_i$ is $a$ when $X_i$ is $1$, and $b$ when $X_i$ is $0$.

I am interested in the behavior of the random variables
$$Z_n = \frac{\sum_{i = 1}^n\left(\prod_{j = 1}^iC_j\right)X_i}{\sum_{i = 1}^n\prod_{j = 1}^i C_j} $$

as $n\to \infty$. Does $Z_n$ converge a.s.? Does $Z_n$ converge in distribution?

Note that the product of the $C_j$'s is sometimes very small and sometimes very large, as the central limit theorem says that $\frac{1}{\sqrt n} \sum \log C_i$ converges in distribution to a normal $\mathcal N (0, \sigma^2)$. This follows from the condition that $a^pb^{1-p} = 1$.

I ran some numerical experiments in Mathematica that suggest that $Z_n$ does converge to a constant, but this constant (presumably the limit of the means of $Z_n$) is a non-trivial function of $a$, $b$, and $p$. Indeed, the mean of $Z_n$ is difficult to compute, and does not seem to simplify nicely.

I am not a probabilist, so any resource that deals with this kind of random variable would be helpful.

1more comment