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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 ♦
    Commented Jun 20, 2020 at 22:46
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    $\begingroup$ Correct -- this would be the only random variable with such moments. $\endgroup$
    – Ricardo Batista
    Commented Jun 20, 2020 at 22:51
  • $\begingroup$ See stats.stackexchange.com/questions/84219/… for non-unicity, but there the mgf do not exist in an open interval around zero $\endgroup$
    – kjetil b halvorsen ♦
    Commented Sep 7, 2023 at 15:48

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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:

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