# 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}
}
• Published 15 May 2017
• Mathematics
• St. Petersburg Mathematical Journal
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… Expand
3 Citations
A note on norms of signed sums of vectors
• Mathematics
• 2019
Abstract Improving a result of Hajela, we show for every function f with limn→∞f(n) = ∞ and f(n) = o(n) that there exists n0 = n0(f) such that for every n ⩾ n0 and any S ⊆ {–1, 1}n with cardinalityExpand
Hypercontractivity, and Lower Deviation Estimates in Normed Spaces
• Mathematics
• 2019
We consider the problem of estimating probabilities of lower deviation $\mathbb P\{\|G\| \leqslant \delta \mathbb E\|G\|\}$ in normed spaces with respect to the Gaussian measure. These estimatesExpand
Quelques inégalités de superconcentration : théorie et applications
Cette these porte sur le phenomene de superconcentration qui apparait dans l'etude des fluctuations de divers modeles de la recherche actuelle (matrices aleatoires, verres de spins, champ libreExpand

#### References

SHOWING 1-10 OF 33 REFERENCES
The Randomized Dvoretzky’s Theorem in $$l_{\infty }^{n}$$ and the χ-Distribution
Let $$\varepsilon \in (0,1/2)$$. We prove that if for some n > 1 and k > 1, a majority of k-dimensional sections of the ball in $$l_{\infty }^{n}$$ is $$(1+\varepsilon )$$-spherical then necessarilyExpand
On Dvoretzky's theorem for subspaces of L
• Mathematics
• Journal of Functional Analysis
• 2018
We prove that for any $2 \varepsilon \mathbb E\|Z\| \right) \leq C \exp \left (- c \min \left\{ \alpha_p \varepsilon^2 n, (\varepsilon n)^{2/p} \right\} \right), \quad 0 0$ is a constant dependingExpand
Upper Bound for the Dvoretzky Dimension in Milman-Schechtman Theorem
• Mathematics
• 2017
For a symmetric convex body $$K \subset \mathbb{R}^{n}$$, the Dvoretzky dimension k(K) is the largest dimension for which a random central section of K is almost spherical. A Dvoretzky-type theoremExpand
The surface measure and cone measure on the sphere of $\ell_p^n$
• Mathematics
• 2006
We prove a concentration inequality for the l n p norm on the l n p sphere for p,q > 0. This inequality, which generalizes results of Schechtman and Zinn (2000), is used to study the distance betweenExpand
Superconcentration, and randomized Dvoretzky's theorem for spaces with 1-unconditional bases
Abstract Let n be a sufficiently large natural number and let B be an origin-symmetric convex body in R n in the l-position, and such that the space ( R n , ‖ ⋅ ‖ B ) admits a 1-unconditional basis.Expand
The Random Version of Dvoretzky's Theorem in l_{\infty}^n
We show that with “high probability” a section of the `∞ ball of dimension k ≤ cε log n (c > 0 a universal constant) is ε close to a multiple of the Euclidean ball in this section. We also show that,Expand
Concentration Inequalities - A Nonasymptotic Theory of Independence
• Mathematics, Computer Science
• Concentration Inequalities
• 2013
Deep connections with isoperimetric problems are revealed whilst special attention is paid to applications to the supremum of empirical processes. Expand
Asymptotic Theory Of Finite Dimensional Normed Spaces
• Mathematics
• 1986
The Concentration of Measure Phenomenon in the Theory of Normed Spaces.- Preliminaries.- The Isoperimetric Inequality on Sn?1 and Some Consequences.- Finite Dimensional Normed Spaces, Preliminaries.-Expand
Random version of Dvoretzky’s theorem in ℓpn
• Mathematics
• 2017
We study the dependence on e in the critical dimension k(n,p,e) for which one can find random sections of the lpn-ball which are (1+e)-spherical. We give lower (and upper) estimates for k(n,p,e) forExpand
Euclidean Sections of Convex Bodies
This is a somewhat expanded form of a 4h course given, with small variations, first at the educational workshop Probabilistic methods in geometry, Bedlewo, Poland, July 6–12, 2008 and a few weeksExpand