Let $X_i$ be an iid sequence of random variables with support in $(0,1)$. I'm looking for references (even just a name) for the following infinite sum random variable:

$$S:=X_1+X_1X_2+X_1X_2X_3+X_1X_2X_3X_4+\cdots.$$

This came up for a waiting-time problem. I can easily calculate the expected value and variance of the above sum, but I'm interested if other people have studied this in literature, specifically if there are asymptotics for limiting distribution. I presume there are also issues when $P(X>1-\epsilon)$ falls off too slowly with increasing $\epsilon$.