# Example of random variable with a unique moment sequence but mgf DNE in a neighborhood of 0 [duplicate]

Do you have an example of a random variable $$X$$ with a unique moment sequence but whose mgf does not exist in a neighborhood of 0?

In other words, I'm looking for a counterexample to the converse of the statement: if $$M_X(t)$$ exists in a neighborhood of 0, then the moments of $$X$$ uniquely determine its distribution.

• Every random variable has a definite moment sequence. By "unique moment sequence" do you mean this would be the only random variable with such moments?
– whuber
Commented Jun 20, 2020 at 22:46
• Correct -- this would be the only random variable with such moments. Commented Jun 20, 2020 at 22:51
• See stats.stackexchange.com/questions/84219/… for non-unicity, but there the mgf do not exist in an open interval around zero Commented Sep 7, 2023 at 15:48

## 1 Answer

You will not find such an example, this is explained at Whether distributions with the same moments are identical. For more information some other posts: