A general approach to problems of this type, worked out for a slightly different continued fraction, 
$$Y_n=X_n+1/X_{n-1},$$
is described in <A HREF="https://www.jstor.org/stable/25055738?seq=1#page_scan_tab_contents">Random Continued Fractions: A Markov Chain Approach</A> (2004). 

A closed-form answer follows if the $X_n$'s have a Gamma distribution, $$P(X)\propto X^{\lambda-1}e^{-aX},\;\; X> 0,$$ 
when the $n\rightarrow\infty$ limit of $Y_n$ tends to the distribution $$P(Y)\propto Y^{\lambda-1}\exp[-a(Y+1/Y)],\;\; Y>0.$$ 
This result goes back to <A HREF="https://link.springer.com/article/10.1007%2FBF00534200?LI=true">A characterization of the generalized inverse Gaussian distribution by continued fractions</A> (1983). A Bernoulli distribution for the $X_n$'s gives a more complicated answer for $P(Y)$.