# Two-sided estimates for order statistics of log-concave random vectors

## TL;DR

This paper derives precise two-sided bounds for the expectations of order statistics of log-concave random vectors, advancing understanding of their coordinate maxima and sums with universal constants.

## Contribution

It provides the first sharp two-sided bounds for order statistics of log-concave vectors, applicable in unconditional and isotropic cases, with explicit bounds for sums of largest coordinates.

## Key findings

- Exact bounds up to universal constants for order statistics in unconditional case.
- Bounds valid for all k in unconditional case and for k ≤ n - c n^{5/6} in isotropic case.
- Estimates for sums of the largest moduli of coordinates for specific classes of vectors.

## Abstract

We establish two-sided bounds for expectations of order statistics ($k$-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all $k$ and in the isotropic case for $k \leq n-cn^{5/6}$. We also derive two-sided estimates for expectations of sums of $k$ largest moduli of coordinates for some classes of random vectors.