Variance of maximum of Gaussian random variables

by Devil   Last Updated May 25, 2019 00:19 AM

Given random variables $X_1,X_2, \cdots, X_n$ sampled iid from $\sim \mathcal{N}(0, \sigma^2)$, define $$Z = \max_{i \in \{1,2,\cdots, n \}} X_i$$

We have that $\mathbb{E}[Z] \le \sigma \sqrt{2 \log n}$. I was wondering if there are any upper/lower bounds on $\text{Var}(Z)$?

Answers 2

You can obtain upper bound by applying Talagrand inequality : look at Chatterjee's book (Superconcentration phenomenon for instance) .

It tells you that ${\rm Var}(f)\leq C\sum_{i=1}^n\frac{\|\partial_if\|_2^2}{1+\log( \|\partial_i f||_2/\|\partial_i f\|_1)}$.

For the maximum, you get $\partial_if=1_{X_i=max}$, then by integrating with respect to the Gaussian measure on $\mathbb{R}^n$ you get $\|\partial_if\|_2^2=\|\partial_if\|_1=\frac{1}{n}$ by symmetry. (Here I choose all my rv iid with variance one).

This the true order of the variance : since you have some upper bound on the expectation of the maximum, this article of Eldan-Ding Zhai (On Multiple peaks and moderate deviation of Gaussian supremum) tells you that
${\rm Var}(\max X_i)\geq C/(1+\mathbb{E}[\max X_i])^2$

It is also possible to obtain sharp concentration inequality reflecting these bound on the variance : you can look at or, for more general gaussian process, at my paper

In full generality it is rather hard to find the right order of magnitude of the variance of a Gaussien supremum since the tools from concentration theory are always suboptimal for the maximum function.

Why do you need these kinds of estimates if I may ask ?

Tanguy Kevin
Tanguy Kevin
January 02, 2017 11:21 AM

More generally, the expectation and variance of the range depends on how fat the tail of your distribution is. For the variance, it is $O(n^{-B})$ where $B$ depends on your distribution ($B = 2$ for uniform, $B = 1$ for Gaussian, and $B = 0$ for exponential.) See here. The table below shows the order of magnitude for the range.

enter image description here

Vincent Granville
Vincent Granville
May 24, 2019 23:33 PM

Related Questions

Improving variable estimates by reducing uncertainty

Updated August 21, 2017 07:19 AM

I need help to show that $E(\sum x)=\sum E(x)$

Updated August 14, 2016 08:08 AM