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Topic: Question: Chi-Square Weighted Linear Sum?
Replies: 1   Last Post: Jun 20, 1996 11:28 AM

 Herman Rubin Posts: 968 Registered: 12/6/04
Re: Question: Chi-Square Weighted Linear Sum?
Posted: Jun 20, 1996 11:28 AM

In article <craigf-1906961009020001@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 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
hrubin@stat.purdue.edu Phone: (317)494-6054 FAX: (317)494-0558