So this is my first post in mathoverflow. I posted this problem in **Mathstack**, an I've also
put a bounty on it, but did not get any response. If anyone can at least point out a reference on this problem, I will be grateful.

Let $X_1,X_2,\dots$ be independent random variables such that $P(X_i=1)=p=1- P(X_i=\epsilon_i)$, for some $0<\epsilon_i<1$, and let

$$Y=X_1+\frac{X_1}{X_2+\frac{X_2}{X_3+\frac{X_3}{X_4+\dots}}} \, .$$

**What is the distribution of $Y$?****What is the characteristic function of $Y$?**

We can think of **less complicated** version:

Let $X_1,X_2,\dots$ be i.i.d. random variables such that $P(X_1=1)=p=1-P(X_1=\epsilon)$ for some $0<\epsilon <1$

$$Y=X_1+\frac{X_1}{X_2+\frac{X_2}{X_3+\frac{X_3}{X_4+\dots}}}$$

Here we can write: $Y=Y_1$, $Y_1=X_1(1+\frac{1}{Y_2})$, $Y_2=X_2(1+\frac{1}{Y_3})$ and so on. It can be easily seen that $Y_i$'s are identical but not independent. So, I think that first we should answer questions **1** and **2** for this simpler case.