Dealing with Recurrence Relations of Random Variables

Let $(Y_n)_{n\in \mathbb N}$ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N}$ another sequence, defined recursively as follows:

$$X_{n+1} = \alpha X_n + \beta Y_n \qquad \forall n \ge 2$$ $$X_1 = Y_1$$

where $\alpha, \beta \in \mathbb R$.

Is there a general approach to dealing with such random variables? If I wanted to determine the moments, or the distribution of $(X_n)$, assuming I know the distribution of $(Y_n)$, are there any methods that can be applied here? What if we restricted $(Y_n)$ to be an iid sequence?

I came across a similar post here, in which the answer makes reference to terminology used in stochastic processes, which I am not familiar with. Are there any other ways of going about this problem, or can anyone shed some more light on what was done in the post I linked? If yes, could any of these methods also be used towards the much more complex problem of analyzing $(X_n)$ if it were defined as follows, for some Borel-measurable function $f: \mathbb R^2 \rightarrow \mathbb R$:

$$X_{n+1} = f(X_n, Y_n) \qquad \forall n \ge 2$$ $$X_1 = Y_1$$

• You'll need to know about the covariances between the $X_i$ and the $Y_i$ as well as between the $Y_i$.
• Obviously, for every $n\geqslant1$, $$X_n=\alpha^{n-1}Y_1+\beta\sum_{k=1}^{n-1}\alpha^{n-k-1}Y_k,$$ and the moments of $X_n$ follow. Determining the distribution is, in the general case, notoriously more difficult.