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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.

Ricardo Batista
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  • 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 '20 at 22:46
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    Correct -- this would be the only random variable with such moments. – Ricardo Batista Jun 20 '20 at 22:51
  • See https://stats.stackexchange.com/questions/84219/whether-distributions-with-the-same-moments-are-identical for non-unicity, but there the mgf do not exist in an open interval around zero – kjetil b halvorsen Sep 07 '23 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:

kjetil b halvorsen
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