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5 votes
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857 views

Anti-concentration inequality for Gaussian random vector

I am trying to obtain an explicit expression for $C$ in terms of $b$ in the following inequality. Suppose that $Y$ is a centred Gaussian random vector in $\mathbb R^p$ such that $\operatorname EY_j^...
Cm7F7Bb's user avatar
  • 423
4 votes
0 answers
190 views

Pedestrian proof of Gaussian chaos for order-two polynomial?

Let $\ell \geqslant 1$. Let us consider $(g_n)_{n \in \mathbb{N}}$ identically distributed independent real gaussian variables and real number $(a_{n_1,\dots n_{\ell}})_{(n_1, \dots, n_{\ell}s)\in\...
combNightmare's user avatar
3 votes
0 answers
79 views

Order of $\mathbb{E}[ \max_i |x_i + z_i| - \max_i |z_i|]$

Let $z_1, \dots, z_n$ be iid standard Normal, and let $x \in \mathbb{R}^n$. Put $\|u\|_\infty = \max_i |u_i|$. Define $$ F(x) = \mathbb{E}\Big[\|x + z\|_\infty - \|z\|_\infty\Big] $$ If $\|x\|_\infty \...
Drew Brady's user avatar
3 votes
0 answers
185 views

Measure change bound for function of subgaussian r.v

Let $X$ be a (sub)gaussian r.v. on $\mathbb{R}^d$; say $X\sim\mathcal{N}(\mathbf{0},\mathbb{1}_d)$; and let $a\colon\mathbb{R}^d\to [0,1]$ be a function with $\mathbb{E}[a(X)] > 0$. It is not hard ...
Clement C.'s user avatar
  • 1,372
2 votes
0 answers
97 views
+100

Inequalities for norm of centered Gaussian and uncentered Gaussian

Let $g$ denote a standard Gaussian vector in $\mathbb{R}^n$, and $\|\cdot\|$ a norm. Let $x \in \mathbb{R}^n$ and define $$ F(x) = \mathbb{E}[\|x + g\| - \|g\|]. $$ I am wondering if it is possible to ...
Drew Brady's user avatar