Re: Multinomial approximation to Poisson ??

Nag wrote:
Think of convergence of Binomial to Poisson. Binomial is 2-variate by
your argument and yet converges to a univariate Poisson. Given
Poisson(L) and n, Binomial(p,L/n) converges to Poisson (L) as n -> oo.

I assume you meant to say "Binomial(n,L/n) converges to Poisson (L)"

Note that the mean of Binomial(n,L/n) is L, and that the variance is
L(1-L/n) which tends towards a fixed L as n increases. By happy
coincidence the mean and variance of Poisson(L) are both equal to L.

Also note the problem that if, instead of fixing L, you fix p and let L
increase with n by having L=np, so the variance of the Binomial becomes
np(1-p) and never gets closer in any sense to the mean np. So you
cannot say "Binomial(n,p) converges to Poisson (np)".


Interpret my problem in the same way. Instead of 0 or 1 in each trial,
we get 0 or 1 or 2 or 3 or 4 in each trial. My interest is in the sum
of outcomes of n such trials and the behavior of this sum as n -> oo.

If the variance of your multinomial is not equal to its mean then a
Poisson approximation cannot help you.

.

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