lp norm of multivariate standard normal random variable

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.

$$\mathbb{P}(||X||_{l^p}\le Cn^{e(p)} )$$ C is allowed to depend on $p$.

Thank you, warsaga

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 Are $X_i$ independent? – Davide Giraudo Nov 4 '12 at 18:14 The case $p=1$ is the easiest :) – Hans Engler Nov 4 '12 at 18:43

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