# Undefined MGF but all moments finite?

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?

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. 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.

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'.