Re: Distribution of random sum of random variables
From: Eric Wong (wongman_eric_at_yahoo.com)
Date: 07/29/04
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Date: 29 Jul 2004 08:48:45 -0700
israel@math.ubc.ca (Robert Israel) wrote in message news:<ce68td$pu5$1@nntp.itservices.ubc.ca>...
> In article <a07a8e30.0407270736.1277df26@posting.google.com>,
> Eric Wong <wongman_eric@yahoo.com> wrote:
> >Suppose that
> >1. N is a discrete nonnegative random variable
> >2. X_i's are i.i.d. random variables
> >2. N and X_i are independent
>
> >Is it true that when E[N] tends to infinity, the distribution of the
> >following random sum converges to the normal distribution? (as I guess
> >it may be an extension of the Central Limit Theorem.)
>
> >S = sum(i=1..N) X_i
>
> I'm guessing that what you mean is something like this:
> Let the distribution of N depend on some parameter t, such that E[N] = t.
> As t -> infinity, does (S - E[S])/sqrt(Var(S)) converge in distribution
> to a normal distribution?
>
> The answer is no in general. Take, for example, a case where
> N = 0 with probability 1/2 and 2t with probability 1/2.
>
> >Furthermore, for the special case that N and X_i are both Poisson, is
> >there some well-known distribution that describes S?
>
> The phrase to look up is "compound Poisson process", I think.
>
> Robert Israel israel@math.ubc.ca
> Department of Mathematics http://www.math.ubc.ca/~israel
> University of British Columbia
> Vancouver, BC, Canada V6T 1Z2
Thanks, Professor. You are right, this is what I meant by "E[N] tends
to infinity". Sorry for my imprecise question.
About your counter example, though the limiting distribution will
never be exactly normal, I think the density function looks almost
normal except at the point zero when t becomes very large, as a result
of the CLT.
I have written some programs in Matlab to explore the distribution
when both N and X_i are Poisson. I found that the distribution becomes
very close to normal when the expectation of N is large compared to
that of X_i. This leads me to guess that it might be a rather general
result. Though I agree it is not true in every case, intuitively I
think the approximation by the normal distribution might hold for a
large class of distributions of N and X_i. As the problem looks quite
interesting, I suppose it must be well studied by someone before.
In case you are interested to know, this problem arose when I tried to
model the number of requests hitting at a web server in a unit time,
where N is the number of users simultaneously using a system, and X_i
is the frequency of requests by each user.
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