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 Random Continued Fractions: A Markov Chain Approach (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 characterization of the generalized inverse Gaussian distribution by continued fractions (1983). A Bernoulli distribution for the $X_n$'s gives a more complicated answer for $P(Y)$.