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I can find the $i th$ term of a recurrence relationship $Y$ like this:

$\begin{equation} Y_i=\sigma_i+\beta_iY_{i-1} \end{equation}$ where $\sigma$ and $\beta$ are normally distributed random variables. I'd like to convert this into summation over $i=1..n$ something like this:

$\begin{equation} Y=\sigma_i+\displaystyle \sum_{i=2}^{n-1}\beta_iY_{i-1} \end{equation}$

I know this summation is wrong, any ideas?

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Start without the $\Sigma$ in there: $Y_i=\sigma_i+\beta_i(\sigma_{i-1}+\beta_{i-1}(\sigma_{i-2}+...))= \sigma_i+\beta_i\sigma_{i-1}+\beta_i\beta_{i-1}\sigma_{i-2}= \sum_{j=0}^i \left(\prod_{k=j+1}^i \beta_k\sigma_j\right)$

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    Yeah, they're meaningless. It's like $Y_i$ is a sum over the first $i+1$ $\sigma$s, which are indexed by $j$, and each $\sigma_j$ has a coefficient that's a product of some $\beta$s, which are indexed by $k$-but $k$ starts over for each $\sigma_j$. Try unwrapping the expression for $Y_1, Y_2$ to see better how it works.2012-10-17