What are the sampling distributions of higher moments of the normal distribution? Let $X_i$ be independent, normally distributed random variables, for $1\leq i\leq N$.  What is the distribution of $Y_m=\frac1 N \sum_{i=1}^N X_i^m$?
Every high school student knows part of the answer.  The mean of $Y^m$ is the $m$'th moment of the normal distribution, and the variance of $Y^1$ is $1/N$.
I'm interested in the width of the distribution of $Y^m$.  How does its variance scale with the sample size $N$?  (Is variance a useful measure?)  I'm especially interested in the large sample limit, with m a small integer.  Knowing where to look it up would be helpful.
 A: 
I'm interested in the width of the distribution of $Y_m$. How does its variance scale with the sample size $N$?

$\text{var}(X_i^m) = E[X_i^{2m}] - (E[X_i^m])^2$ is easily evaluated from the moment-generating function $\exp(\sigma^2t^2/2 + \mu t)$ of the $N(\mu, \sigma^2)$ random variable $X_i$ (or from the characteristic function as suggested in the comment by mpiktas).  Since this variance does not depend on $i$, 
let us denote it by $\text{var}(X^m)$.  Then, since the $X_i^m$'s are independent random variables (they are functions of independent random variables), we have
$$\text{var}(Y_m) = \text{var}\left(\frac{1}{N}\sum_{i=1}^N X_i^m\right)
= \frac{1}{N^2}\left(\sum_{i=1}^N \text{var}(X_i^m)\right)
= \frac{1}{N}\text{var}(X^m)$$
Note that the displayed equation above does not require that the $X_i$ be 
normal random variables.  For i.i.d. random variables $Z_i$ (with finite 
second moment), the 
variance of $N^{-1}\sum_i Z_i$, the average of $N$ variables, is always
$N^{-1}\text{var}(Z)$, that is the variance always scales as $1/N$.
For general distributions, the Chebyshev inequality can be used to obtain
a weak bound on the width of the distribution.  See here for a related discussion. 
