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Small cover approach to the suprema of positive canonical processes

Witold Bednorz, Rafał Martynek, Rafał Meller

TL;DR

This work develops a small-cover framework to bound the expected supremum of positive canonical processes, extending Park–Pham’s results from a positive selector setting to canonical processes driven by iid nonnegative variables under mild tail assumptions and to general iid positive variables. It constructs witness-based small covers that certify largeness of the supremum via a union bound over coordinate-threshold events, yielding both discrete and iid variants and enabling a new comparison principle for processes on generalized Orlicz balls. The results lead to Sudakov-type bounds in dependent settings, and they provide discrete-to-continuous transfers that connect to classical chaining techniques while leveraging nonnegativity. Overall, the small-cover approach offers a versatile tool for analyzing suprema of nonnegative canonical processes with broad tail structures and dependencies, with potential applications in Sudakov-type bounds and related concentration phenomena.

Abstract

We extend the recent result of Park and Pham concerning the positive selector process to canonical processes generated by i.i.d. nonnegative random variables satisfying minimal tail assumptions. We also provide a result of the same nature for canonical processes based on general i.i.d. positive variables.

Small cover approach to the suprema of positive canonical processes

TL;DR

This work develops a small-cover framework to bound the expected supremum of positive canonical processes, extending Park–Pham’s results from a positive selector setting to canonical processes driven by iid nonnegative variables under mild tail assumptions and to general iid positive variables. It constructs witness-based small covers that certify largeness of the supremum via a union bound over coordinate-threshold events, yielding both discrete and iid variants and enabling a new comparison principle for processes on generalized Orlicz balls. The results lead to Sudakov-type bounds in dependent settings, and they provide discrete-to-continuous transfers that connect to classical chaining techniques while leveraging nonnegativity. Overall, the small-cover approach offers a versatile tool for analyzing suprema of nonnegative canonical processes with broad tail structures and dependencies, with potential applications in Sudakov-type bounds and related concentration phenomena.

Abstract

We extend the recent result of Park and Pham concerning the positive selector process to canonical processes generated by i.i.d. nonnegative random variables satisfying minimal tail assumptions. We also provide a result of the same nature for canonical processes based on general i.i.d. positive variables.
Paper Structure (5 sections, 16 theorems, 167 equations)

This paper contains 5 sections, 16 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Application and Proofs
  4. Proof of Theorem \ref{['thm:wykl']}
  5. Proof of Theorem \ref{['thm:rozdconentracja']}

Key Result

Theorem 2.1

Fix $\delta\in (0,1)$, $K\geq 1$ and $T\subset {\mathbb R}_0^d$. Let $X_1,\ldots,X_d$ be i.i.d. non-negative random variables such that for some $C>1$ we have, Assume that $f_1,\ldots,f_d$ are strictly increasing, nonnegative functions defined on ${\mathbb R}_0$ such that for each $i\leq d$ for some $C'>1$. Then there exists a constant $L=L(\delta,C,C',K)$ and a family ${\cal G}\subset \{(x_\ast,

Theorems & Definitions (48)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4: Comparison principle for generalized Orlicz balls
  • Remark 2.5
  • Theorem 2.6
  • proof
  • proof
  • Lemma 3.3
  • proof
  • ...and 38 more