Undefined MGF but all moments finite?

Question

For the lognormal distribution:

https://en.wikipedia.org/wiki/Log-normal_distribution

The moment generating function is undefined, but all the moments exist and are finite.

I thought the moment generating function characterizes the moments of a distribution, but there are cases of distributions with finite moments but undefined MGF. How can one explain this?

This might be useful: https://stats.stackexchange.com/q/389846/89012 — SOULed_Outt, Apr 26 '20 at 18:00
@SOULed_Outt Thanks. As usual, the legendary cardinal providing amazing answers. — MGF, Apr 26 '20 at 18:05
See also https://stats.stackexchange.com/questions/32706/existence-of-the-moment-generating-function-and-variance for fact about mgf's — kjetil b halvorsen, Aug 25 '21 at 00:15

score 1 · Answer 1 · edited Jun 11 '20 at 14:32

Per Wikipedia on moment-generating functions to quote:

However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the characteristic function or Fourier transform always exists (because it is the integral of a bounded function on a space of finite measure), and for some purposes may be used instead.

In particular, for the log-normal distribution:

All moments of the log-normal distribution exist...However, the expected value ${E} {[e^{tX}]}$ is not defined for any positive value of the argument t as the defining integral diverges. In consequence the moment generating function is not defined.[5] The last is related to the fact that the lognormal distribution is not uniquely determined by its moments.

The characteristic function ${E} {[e^{itX}]}$ is defined for real values of t but is not defined for any complex value of t that has a negative imaginary part, and therefore the characteristic function is not analytic at the origin. In consequence, the characteristic function of the log-normal distribution cannot be represented as an infinite convergent series.[6]

So, even though all moments of the log-normal distribution exist, the MGF defining integral must still converge for positive and negative values of t. Interestingly, the characteristic function of the log-normal distribution also does not produce a convergent series either for t a complex number with a negative imaginary number.

This may be related to the apparent fact that 'log-normal distribution is not uniquely determined by its moments'.

Undefined MGF but all moments finite?

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