M asked, what happens when the variance of a distribution is equal to zero, can it still have higher moments different from zero?

Let’s use the Cauchy–Schwarz inequality:

\[\begin{align*} |\int_{\mathbb{R}^{d}}f(x)\overline{g(x)}dx|^2\leq & \int_{\mathbb{R}^{d}}|f(x)|^2dx \int_{\mathbb{R}^{d}}|g(x)|^2dx \end{align*}\]

if we are calculating the $n$ moment of $p(x)$

\[\begin{align*} |\int_{\mathbb{R}^{d}}x^np(x)dx|^2 \leq & \int_{\mathbb{R}^{d}}|xp(x)^{1/2}|^2dx \int_{\mathbb{R}^{d}}|x^{n-1}p(x)^{1/2}|^2dx \\ = & \int_{\mathbb{R}^{d}}x^2p(x)dx \int_{\mathbb{R}^{d}}x^{2(n-1)}p(x)dx \end{align*}\]

Same reasoning applies for centered moments, only change $x^n$ to $(x-x_0)^n$. So, to answer to M: if the variance is zero, all higher moments will be zero as well. Interesting to think that you can always bound a moment with other two, even if actually it’s already bounded by the integral of $|x^n|$. Like the third moment is bounded by the sqrt of the multiplication of the second and the fourth.

For a gaussian central moments e.g. $E[x^3]\leq\sqrt{E[x^2]E[x^4]}$ is satisfied trivially since $0\leq\sqrt{\sigma^23\sigma^4}$, and $E[x^4]\leq\sqrt{E[x^2]E[x^6]}$ becomes $3\sigma^4\leq\sqrt{\sigma^215\sigma^6} = \sqrt{15}\sigma^4 =3.872\cdots\sigma^4$ which is tightier than I was expecting.

For a gamma distribution uncentered moments $E[x^3]\leq\sqrt{E[x^2]E[x^4]}$ becomes $\frac{\alpha(\alpha+1)(\alpha+2)}{\lambda^3}\leq\sqrt{\frac{\alpha(\alpha+1)}{\lambda^2}\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)}{\lambda^4}}$ which after some cancellations becomes $\sqrt{\alpha+2}\leq\sqrt{\alpha+3}$ or $1\geq0$, which is true as well.

This result applies for general distribution, but it will become quite useless for example for distributions that don’t have a converging second moment, such as the Cauchy distribution.

It is interesting to think that odd moments are bounded by even moments, but not necessarily the other way around. But probably it is not true, and the general form of the Cauchy Swartz inequality, where you don’t use anymore $L_2$, but $L_q$, could be used to prove as well a bound on even moments with odd moments.

A minimal generalization is given by what I will call Theorem 2

\[\begin{align*} |\int_{\mathbb{R}^{d}}x^np(x)dx|^2 \leq & \int_{\mathbb{R}^{d}}x^{2m}p(x)dx \int_{\mathbb{R}^{d}} x^{2n-2m}p(x)dx \\ E[x^n]\leq & \sqrt{E[x^{2m}]E[x^{2(n-m)}]} \end{align*}\]

where $m$ can be any number, integer, real or complex.