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Re: Question: ChiSquare Weighted Linear Sum?
Posted:
Jun 20, 1996 11:28 AM


In article <craigf1906961009020001@dialup44.afn.org>, Craig Fancourt <craigf@grove.ufl.edu> 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 chisquare random >variable as follows:
>y(n) = x2 + g * y(n1)
>where x2 is a first order chisquare random variable and g >is a deterministic feedback parameter (0<g<=1). This can be >viewed as a linear sum of weighted first order chisquare >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 >gammalike 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(nk).
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 IN479071399 hrubin@stat.purdue.edu Phone: (317)4946054 FAX: (317)4940558



