Concentration estimates for functions of finite high-dimensional random arrays

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.