Properties of Moment Functions I was watching this video here https://www.youtube.com/watch?v=ZLJqjiI0aHM/
From 5:00 - 5:20, the author of this video states that the "variance of higher order moments tends to increase".
However, no real explanation in the video is provided as to why this is.
I tried to search online but I could not find any explanation behind this.
For instance, how can I be mathematically certain that the variance of higher order moments does in fact increase? Is there some mathematical proof for this?
Thanks!
Note: The only reason (non mathematical) I could think of is that since the Variance is "additive" - perhaps higher order moments have more terms and functions with more terms will likely have larger variances ... but I am not sure if this is the case.
 A: The statement is simply not true, formally. For the sake of simplicity, consider some finitely supported distribution, with support $\mathcal S$.
Assume that $\mathcal S\subseteq(0,1)$. Notice that $X^n\le X^m$ for $n\ge m$, meaning the values will (deterministically) decrease when we look at higher moments. Put simply, for any $\varepsilon>0$ we find $N$ such that for all $n\ge N$ the support $\mathcal S_n$ of $X^n$ is in $\mathcal S^n\subseteq(0,\varepsilon)$. This means that $\mathbb E[X^n]\in(0,\varepsilon)$ and hence $|X^n-\mathbb E[X^n]|<\varepsilon$ almost surely, so the variance is bounded by $\varepsilon^2$, arbitrarily small. This does not grow.
However, we may deal with numbers greater $1$. So, assuming that $x=\max\mathcal S>1$ and that $y=\max\mathcal S\setminus\{x\}$ is well-defined (i.e. $|\mathcal S|>1$, still with $\min\mathcal S>0$) we have $\mathbb E[X^n]\le(1-p)y^n+px^n=((1-p)(y/x)^n+p)x^n$, where $p=\mathbb P(X=x)$. Since $(y/x)^n$ tends to $0$, the expectation is asymptotically bounded by $px^n$ up to some small relative error, $(1+o(1))px^n$ in Landau notation. This suggests that $x^n-\mathbb E[X^n]\ge(1-p)x^n$ (with some small relative error), which tends to $\infty$. This shows that the variance explodes as soon as you have a value in the (non-trivial) support that is greater than $1$ (for a non-negative finitely supported random variable). However, in order to understand the behaviour of moments, this is the relevant concept.
