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?