# 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.

• Did you consider using Central Limit Theorem? $X_i^m$ will be iid., so $\sqrt{N}(Y^{(m)}-EX_1^m)\to N(0,var(X_1^m))$. You can find $EX_1^m$ and $Var(X_1^m)=EX_1^{2m}-(EX_1^m)^2$ by differencing the characteristic function of $X_1$ and evaluating it at zero, which for normal variables is not that hard to do. As a bonus, all the higher moments will be expressed as the functions of the first and second moment. Oct 31, 2011 at 13:01
• Why is the variance of Y $1/(n-1)$ ? Is X standard normal? And shouldn't it be $n$ in the denominator? Oct 31, 2011 at 17:04
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.
• Recently, the TeX formatting in many posts seems to have gone a bit weird. I tried to edit this post to make $\text into $ \text which would apparently have fixed the problem but singe-space edits are not allowed. - at least I'm assuming that this isn't just an error on my end but happens everywhere. Apr 15, 2014 at 10:55