Distribution of random sum of random variables

From: Eric Wong (wongman_eric_at_yahoo.com)
Date: 07/27/04 

Date: 27 Jul 2004 08:36:33 -0700

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

Furthermore, for the special case that N and X_i are both Poisson, is
there some well-known distribution that describes S?

Grateful if someone can point me to some reference books/papers that
cover this type of problem.

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