In article <firstname.lastname@example.org>, Craig Fancourt <email@example.com> wrote: >Hi,
>I've been working on this problem for a week with no luck. >Any help or insight would be greatly appreciated.
>I am recursively filtering a first order chi-square random >variable as follows:
>y(n) = x2 + g * y(n-1)
>where x2 is a first order chi-square random variable and g >is a deterministic feedback parameter (0<g<=1). This can be >viewed as a linear sum of weighted first order chi-square >random variables.
>My question is this:
>What will be the theoretical distribution of y in the limit as >n->infinity?
>I have experimentally determined that the distribution is >gamma-like as g->1 and g->0, but deviates at intermediate values.
I would be surprised if this distribution has been studied. The limiting distribution is the distribution of
\sum g^k * x2(n-k).
The logarithem of the moment generating function is
s(t) = -h * \sum ln(1 - c*g^k*t),
The distribution is clearly approximately the distribution of x2 as g -> 0, but for g -> 1, the Central Limit Theorem comes in and it is asymptotically normal. -- Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399 firstname.lastname@example.org Phone: (317)494-6054 FAX: (317)494-0558