moment generating function an infinite moments

Asked Nov 15 '20 at 14:06

Active Nov 15 '20 at 21:28

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I consider a random variable which has moments $E[X^n]= \infty$. Does this imply that the moment generating function does not exists?

How could I prove that?

The taylor series of mgf is: $E(e^{tX})= 1+ E(X) t + \frac{E(X^2) t^2}{2!} + ...$

This holds only for $t \in (-t_0,t_0)$ and $t_0>0$, where $E(e^{tX})< \infty$ see:

If the moments are infinite can I conclude that mgf is infinite using the taylor series?

edited Nov 15 '20 at 21:28

RobPratt

asked Nov 15 '20 at 14:06

Tim

2

yes, moment generating function does not exist. – Surb Nov 15 '20 at 14:12
How can I show that? – Tim Nov 15 '20 at 14:16
This is just logical $M_X(t)=\sum_{n=0}^\infty \frac{\mathbb E[X^n]}{n!}t^n.$ So, if $\mathbb E[X^n]=\infty $, the sum is not well defined. – Surb Nov 15 '20 at 14:18
But you use the fact that $E[e^{tX}]$ is finite for some t – Tim Nov 15 '20 at 14:25
See here: https://math.stackexchange.com/questions/3508569/expected-value-and-swapping-integral-and-sum – Tim Nov 15 '20 at 14:25
Is this not a contradiction? – Tim Nov 15 '20 at 14:26

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