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Decompositions of finite high-dimensional random arrays

Pandelis Dodos, Konstantinos Tyros, Petros Valettas

TL;DR

The 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.

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.

Decompositions of finite high-dimensional random arrays

TL;DR

The 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.

Abstract

A -dimensional random array on a nonempty set is a stochastic process indexed by the set of all -element subsets of . 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.

Paper Structure

This paper contains 48 sections, 23 theorems, 206 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Notions of symmetry
  4. Random arrays with finite-valued entries
  5. Random arrays with square-integrable entries
  6. Outline of the proofs/Structure of the paper
  7. Approximation by a random array of lower complexity
  8. The $\sigma$-algebras $\Sigma(\mathcal{G}^s_\ell,\boldsymbol{X})$
  9. The approximation
  10. Step 1
  11. Step 2
  12. Step 3
  13. Completion of the proof
  14. A coding for distributions
  15. Box norms
  16. ...and 33 more sections

Key Result

Proposition 1.3

For every triple $m,n,d$ of positive integers with $n\geqslant d$, and every $\eta>0$, there exists an integer $N\geqslant n$ with the following property. If $\mathcal{X}$ is a set with $|\mathcal{X}|=m$ and $\boldsymbol{X}$ is an $\mathcal{X}$-valued, $d$-dimensional random array on a set $I$ with

Figures (3)

  • Figure 1: Aligned pairs of partial maps.
  • Figure 2: The set $\mathcal{R}^x_\ell$.
  • Figure 3: The sets $D_i$, $\Gamma_{i,p}$, $\Theta_{i,p,j}$ and $H_{i,p,j,r}$.

Theorems & Definitions (45)

  • Definition 1.1: Random arrays, and their subarrays
  • Definition 1.2: Approximate spreadability
  • Proposition 1.3
  • Theorem 1.4: Distributional decomposition
  • Theorem 1.5
  • Remark 1.6
  • Theorem 1.7: Physical decomposition
  • Proposition 2.1
  • Lemma 2.2: Shift invariance of projections
  • proof
  • ...and 35 more