= Integrability threshold for exponential powers of a Gaussian
{title2=$\mathbb E[Ze^{\alpha Z^{2m}}]$}
Let $Z$ have <standard normal distribution>, let $m\ge1$ be an integer and let $\alpha\in\mathbb R$. A finite <expected value> of $Ze^{\alpha Z^{2m}}$ requires absolute <integrability>, which reduces to finiteness of
$$
\int_0^\infty z\exp(\alpha z^{2m}-z^2/2)\,dz.
$$
For $m=1$ it is finite exactly when $\alpha<1/2$. For every $m\ge2$ it is finite exactly when $\alpha\le0$: the higher power dominates the negative quadratic when $\alpha>0$, and for $\alpha\le0$ the <integrand> is bounded by $ze^{-z^2/2}$. In the finite cases the signed <expected value> is zero by <symmetry>. Otherwise each signed part has infinite <integral>; a formal symmetric cancellation is only a <Cauchy principal value>.
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