Decompositions of finite high-dimensional random arrays

TL;DR

This paper introduces new structural decompositions for finite high-dimensional random arrays with symmetry properties, extending existing theories and providing effective proofs and applications in concentration inequalities.

Contribution

It presents the first finite, spreadable array decompositions generalizing infinitary results and an analogue of Hoeffding/Efron–Stein decomposition for high-dimensional arrays.

Findings

Decomposition of spreadable arrays in finite settings.

Effective proofs for the decompositions.

Applications to concentration of measure.

Abstract

A $d$-dimensional random array on a nonempty set $I$ is a stochastic process $\boldsymbol{X}=\langle X_s:s\in \binom{I}{d}\rangle$ indexed by the set $\binom{I}{d}$ of all $d$-element subsets of $I$. We obtain structural decompositions of finite, high-dimensional random arrays whose distribution is invariant under certain symmetries. Our first main result is a distributional decomposition of finite, (approximately) spreadable, high-dimensional random arrays whose entries take values in a finite set; the two-dimensional case of this result is the finite version of an infinitary decomposition due to Fremlin and Talagrand. Our second main result is a physical decomposition of finite, spreadable, high-dimensional random arrays with square-integrable entries that is the analogue of the Hoeffding/Efron--Stein decomposition. All proofs are effective. We also present applications of these decompositions in the study of concentration of functions of finite, high-dimensional random arrays.