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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.

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

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