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On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes

Jinyoung Park, Huy Tuan Pham

TL;DR

The paper extends Talagrand’s Gaussian tail-covering paradigm to positive selector and positive empirical processes. It introduces the p-small cover framework via upsets and witnesses, and proves a Bernoulli-p analogue (Theorem Conj 5.7) as well as a positive empirical processes analogue (Theorem thm:pos-emp), both ensuring tail events can be controlled by a cheap collection of simple witnesses. The authors develop a multiplicity-enabled version (Theorem thm:multi-w) and show how discretization and multiset techniques bridge from selector processes to general positive empirical processes. The results illuminate the structure of large deviation events in nonnegative settings and tie into broader questions such as the Kahn-Kalai conjecture, while leaving the fully abstract Talagrand06 framework open. The methods center on fragment-based coverings, minimum fragments, and careful combinatorial cost analysis to yield covers with total probability bounded by 1/2.

Abstract

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli-$p$ setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.

On a conjecture of Talagrand on selector processes and a consequence on positive empirical processes

TL;DR

The paper extends Talagrand’s Gaussian tail-covering paradigm to positive selector and positive empirical processes. It introduces the p-small cover framework via upsets and witnesses, and proves a Bernoulli-p analogue (Theorem Conj 5.7) as well as a positive empirical processes analogue (Theorem thm:pos-emp), both ensuring tail events can be controlled by a cheap collection of simple witnesses. The authors develop a multiplicity-enabled version (Theorem thm:multi-w) and show how discretization and multiset techniques bridge from selector processes to general positive empirical processes. The results illuminate the structure of large deviation events in nonnegative settings and tie into broader questions such as the Kahn-Kalai conjecture, while leaving the fully abstract Talagrand06 framework open. The methods center on fragment-based coverings, minimum fragments, and careful combinatorial cost analysis to yield covers with total probability bounded by 1/2.

Abstract

For appropriate Gaussian processes, as a corollary of the majorizing measure theorem, Michel Talagrand (1987) proved that the event that the supremum is significantly larger than its expectation can be covered by a set of half-spaces whose sum of measures is small. We prove a conjecture of Talagrand that is the analog of this result in the Bernoulli- setting, and answer a question of Talagrand on the analogous result for general positive empirical processes.
Paper Structure (8 sections, 16 theorems, 132 equations)

This paper contains 8 sections, 16 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Conj 5.7']}
  3. Reductions between Theorem \ref{['Conj 5.7']}, Theorem \ref{["Conj 5.7'"]} and Theorem \ref{['Conj 5.7"']}
  4. Proof of Theorem \ref{['Conj 5.7"']}
  5. Proof of Theorem \ref{['thm:pos-emp']}
  6. A version of Theorem \ref{['Conj 5.7']} with multiplicities
  7. Proof of Theorem \ref{['thm:pos-emp']} assuming Theorem \ref{['thm:multi-w']}
  8. Proof of Theorem \ref{['thm:multi-w']}

Key Result

Theorem 1.1

There exists $L>0$ such that the following holds. Let $g$ be an $M$-dimensional standard Gaussian vector. For $\mathcal{T} \subseteq \mathbb{R}^M$, consider the process $Z_t = \langle t, g\rangle$ for $t\in \mathcal{T}$. Then one can find a sequence of half-spaces $H_k$ of $\mathbb{R}^M$ with and

Theorems & Definitions (39)

  • Theorem 1.1: Talagrand, Theorem 2.12.2 in Talagrand2
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • proof : Proof of Theorem \ref{['Conj 5.7']} from Theorem \ref{["Conj 5.7'"]}
  • proof : Proof of Theorem \ref{["Conj 5.7'"]} from Theorem \ref{['Conj 5.7"']}
  • proof : Justification.
  • proof : Justification.
  • Definition 2.3
  • ...and 29 more