$L^p$ norm of multivariate standard normal random variable

Question

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 $p$.

Answer 1

Hint: Assuming $X_i$'s are independent you can apply the Central Limit Theorem on $$\frac{1}{n}\left\Vert X\right\Vert_{\ell^p}^p=\frac{1}{n}\sum_{i=1}^n\lvert X_i\rvert ^p.$$