I would like to show that the mgf of lognormal random variable does not exist. This essentially comes down to showing that the following integral diverges:
$$ \int_0^\infty \frac{1}{\sqrt{2\pi}x} e^{xt} e^{-\frac{(\log x)^2}{2}}\,dx$$
By letting $\log x =u$ and after some algebra, I get the following integral (ignoring the constant)
$$\int_{\mathbb{R}} \exp\left( e^u t – \frac{u^2}{2}\right) \,du > \int_0^\infty \exp\left( e^u t – \frac{u^2}{2} \right)\,du$$
It seems somewhat obvious that the integral cannot possibly converge since $e^u$ will "dominate" $u^2$, but how do I show this rigorously? It seems quite difficult to find an antiderivative directly.
Best Answer
Prove $f:=te^u-\frac{u^2}{2}$ is strictly increasing for sufficiently large $u$. Thus $e^f$ is too. Its integral on $u>M$ for $M$ large is then bounded below by the integral of $e^{f(M)}$, which diverges as the integration range is infinite.