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 $p$.