Re: Distribution of random sum of random variables

From: Robert Israel (israel_at_math.ubc.ca)
Date: 07/27/04 

Date: 27 Jul 2004 19:03:09 GMT

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

Relevant Pages

  • Re: Normal Random Variable Generator
    ... Excel, PowerPoint, and VBA add-ins, tutorials ... > random variables in Excel: ... > differ from a normal distribution. ... but I don't know how to use in my VBA code. ...
    (microsoft.public.excel.programming)
  • Re: Distribution of random sum of random variables
    ... N is a discrete nonnegative random variable ... >following random sum converges to the normal distribution? ... When EN is inifinite, the sum is ...
    (sci.math)
  • Re: Statistical Indep and Corr of Normal RVs
    ... uncorrelatedness implies independence is when the random variables ... Suppose two random variables X and Y are jointly normally ... has a multivariate normal distribution. ... "But ZERO correlation implies independence IF and ONLY IF the random ...
    (sci.stat.edu)
  • Re: jointly gaussian
    ... variables with normal distribution? ... if two random variables have a bivariate normal ... point leads me to suspect that you mean something else by the ... phrase "independent of each other" ...
    (comp.dsp)
  • Re: jointly gaussian
    ... quality and helpful! ... If one comes across a jointly normal distribution, ... if two random variables have a bivariate normal ...
    (comp.dsp)