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Let X and Y be two random variables. Denote by $F_X(x)$ and $F_Y(y)$ their CDFs and by $M_X(t)$ and $M_Y(t)$ their MGFs.

It is known that X and Y have the same CDF iff they have the same MGF.

My question is, if $\forall t>0, M_X(t) \le M_Y(t)$, does it imply a direct inequality for the tails of $F_X(x)$ and $F_Y(y)$? (By direct I mean, not via the Laplace transform, because that only gives bounds for both distributions.)

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    $\begingroup$ You are assuming that the Laplace transform of $d(F_Y-F_X)$ is non-negative pointwise. If $X,Y\ge 0$, then these measures are characterized by Bernstein's theorem: en.wikipedia.org/wiki/… $\endgroup$
    – Christian Remling
    Commented Apr 10, 2015 at 1:08
  • $\begingroup$ Without some assumption on the growth of the moments of a probability measure, you cannot conclude that they uniquely determine the measure. Widder's book on Laplace transform gives examples of such non-uniqueness phenomena. $\endgroup$
    – Liviu Nicolaescu
    Commented Apr 10, 2015 at 8:50

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