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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?

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RSolveis the function you're looking for.

The second example evaluates to

Z = RSolveValue[{z[i + 1] == b z[i] + (1 - b) y[i], z[1] == 0}, z , i]
(*Function[{i}, b^(-1 + i) (\!\(\*UnderoverscriptBox[\(\[Sum]\), \(K[1] = 0\), \(\(-1\) +i\)]\(-\((\((\(-1\) + b)\)\ \*SuperscriptBox[\(b\), \(-K[1]\)]\ y[K[1]])\)\)\) -y[0] + b y[0])]*)
Z[1]//Simplify
(*0*)

addendum

The two difference equations are solved with

{Y, Z} = RSolveValue[{
y[i] == a y[i - 1] + (1 - a) x[i], y[0] == 0, 
z[i + 1] == b z[i] + (1 - b) y[i], z[1] == 0}
, {y, z }, i] //Simplify
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  • $\begingroup$ Great - but how would I ask Mathematica to find Z as a function of both y and x? $\endgroup$ – hydrologist Mar 3 at 10:44
  • 1
    $\begingroup$ @hydrologist Both equations might be solved in one step, see my modified answer. $\endgroup$ – Ulrich Neumann Mar 3 at 11:05

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