Concentration estimates for functions of finite high-dimensional random arrays

TL;DR

This paper establishes conditions under which functions of high-dimensional finite random arrays exhibit concentration phenomena, even with dependent entries, with applications in combinatorics and optimality analysis.

Contribution

It provides easily verifiable conditions ensuring concentration of functions of complex dependent arrays, extending previous results to broader classes of random arrays.

Findings

Conditions guarantee concentration after conditioning on large subarrays

Results apply to arrays with dependent entries

Examples demonstrate the optimality of the conditions

Abstract

Let $\boldsymbol{X}$ be a $d$-dimensional random array on $[n]$ whose entries take values in a finite set $\mathcal{X}$, that is, $\boldsymbol{X}=\langle X_s:s\in \binom{[n]}{d}\rangle$ is an $\mathcal{X}$-valued stochastic process indexed by the set $\binom{[n]}{d}$ of all $d$-element subsets of $[n]:=\{1,\dots,n\}$. We give easily checked conditions on $\boldsymbol{X}$ that ensure, for instance, that for every function $f\colon \mathcal{X}^{\binom{[n]}{d}}\to\mathbb{R}$ that satisfies $\mathbb{E}[f(\boldsymbol{X})]=0$ and $\|f(\boldsymbol{X})\|_{L_p}=1$ for some $p>1$, the random variable $f(\boldsymbol{X})$ becomes concentrated after conditioning it on a large subarray of $\boldsymbol{X}$. These conditions cover several classes of random arrays with not necessarily independent entries. Applications are given in combinatorics, and examples are also presented that show the optimality of various aspects of the results.