Reference request: the Gaussian is determined by its moments

Question

It came up in a paper I am coauthoring that a Gaussian distribution is determined by its moments (and in particular that a probability distribution we are considering converges weakly to the Gaussian).

I assume that this fact is extremely well known, and covered in multiple books, but I don't personally know of a good reference. Could someone please recommend one?

Isn't it a consequence that the moment generating function exists everywhere? — Davide Giraudo, Aug 26 '13 at 21:28

score 4 · Accepted Answer · answered Aug 26 '13 at 21:46

All distributions that satisfy "Carleman's condition" are "uniquely determined" by their moments - this is why if two random variables have the same moment generating function, they have the same distribution. The Gaussian distribution is one of them, and is also one of the "stable" distributions, to which properly standardized sums / functions of random variables converge.

There is "The Handbook of the Normal Distribution" by J.K. Patel & C.B. Read (1982) which is a focused reference for the Normal(i.e. Gaussian) distribution, as its title signals.

Cool, will have a look, thank you! – academic Sep 10 '13 at 00:37 — academic, Sep 10 '13 at 00:37

Reference request: the Gaussian is determined by its moments

1 Answers1

Linked