TITLE: $L^p$ norm of multivariate standard normal random variable
QUESTION [5 upvotes]: Given $X_i\sim \mathcal{N}(0,1)$ what is the behaviour of 
$$ ||X||_{l^p}=(\sum_{i=1}^n|X_i|^p )^{1/p}$$
as $n\rightarrow \infty$?
For $p=2$ results about $\chi$-distribution tell us that 
$$\mathbb{P}(||X||_{l^2}\le 2n^\frac{1}{2} )\rightarrow 1.$$
I am interested in analgous statments for $p\ne1$,i.e.
$$\mathbb{P}(||X||_{l^p}\le Cn^{e(p)} ),$$
where $C$ is allowed to depend on $p$.

REPLY [2 votes]: By the law of large numbers for i.i.d. random variables,
$$
\frac1n\sum_{i=1}^n|X_i|^p\to E[|X_1|^p],
$$
almost surely, hence
$$
\frac1{n^{1/p}}\|X\|_p\to E[|X_1|^p]^{1/p},
$$
almost surely. In particular, for every positive $x$,
$$
P[(1-x)n^{1/p}E[|X_1|^p]^{1/p}\leqslant\|X\|_p\leqslant(1+x)n^{1/p}E[|X_1|^p]^{1/p}]\to1.
$$