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

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 Jun 20 at 22:46
• Correct -- this would be the only random variable with such moments. – Ricardo Batista Jun 20 at 22:51