The variance of the $\ell _p^n$-norm of the Gaussian vector, and Dvoretzky’s theorem

@article{Lytova2019TheVO,
  title={The variance of the \$\ell \_p^n\$-norm of the Gaussian vector, and Dvoretzky’s theorem},
  author={Anna Lytova and Konstantin E. Tikhomirov},
  journal={St. Petersburg Mathematical Journal},
  year={2019}
}
Let $n$ be a large integer, and let $G$ be the standard Gaussian vector in $R^n$. Paouris, Valettas and Zinn (2015) showed that for all $p\in[1,c\log n]$, the variance of the $\ell_p^n$--norm of $G$ is equivalent, up to a constant multiple, to $\frac{2^p}{p}n^{2/p-1}$, and for $p\in[C\log n,\infty]$, $\mathbb{Var}\|G\|_p\simeq (\log n)^{-1}$. Here, $C,c>0$ are universal constants. That result left open the question of estimating the variance for $p$ logarithmic in $n$. In this note, we resolve… 
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