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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 $p$.

  • 0
    Are $X_i$ independent?2012-11-04
  • 0
    The case $p=1$ is the easiest :)2012-11-04

2 Answers 2