# recursion – How to solve for multiple (cascading) recurrence relations

I am working with recurrence relations with a simple base:

$$begin{equation} Y_i = aY_{i-1} + (1-a)X_i quad mbox{and if Y_0=0 then} quad Y_n = (1-a)sum_{i=1}^n a^{i-1} X_i tag{1} end{equation}$$

here, $$X_i$$ are identically-distributed independent random variables (although I do not want to specify a distribution type). I am then interested to characterise the series $$Y_n$$ (its moments, correlation structure, etc.) in terms of those of $$X$$ and the parameter $$a$$.

I have done this by hand for (1) and have obtained relationships between $$mathbf{E}(X_n)$$ and $$mathbf{E}(Y_n)$$, $$mathbf{Var}(X_n)$$ and $$mathbf{Var}(Y_n)$$, $$mathbf{skew}(X_n)$$ and $$mathbf{skew}(Y_n)$$, $$mathbf{Kurt}(X_n)$$ and $$mathbf{Kurt}(Y_n)$$ and so on, up to order 6 moments. The algebra becomes rather tedious, but just about manageable.

However, I am now interested to feed this recurrence through a subsequent equation:

$$begin{equation} Z_{j} = bZ_{j-1} + (1-b)Y_j quad mbox{and if Z_0=0 then} quad Z_n = (1-b)sum_{j=1}^n b^{j-1} Y_j tag{2} end{equation}$$

and then find its moments. Having done that, I want to do it again.

My question is this: how (can ?) I use Mathematica to help me do this?