Decompositions of finite high-dimensional random arrays

Abstract

A $d$-dimensional random array on a nonempty set $I$ is a stochastic process $\mathbf{X}=⟨X_s:s\in \left(\genfrac{}{}{0ex}{}{I}{d}\right)⟩$ indexed by the set $\left(\genfrac{}{}{0ex}{}{I}{d}\right)$ 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.