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From the oracle maximal inequality to martingale random fields via finite approximation from below

Yoichi Nishiyama

Abstract

A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal entropy-based methods. The proposed approach begins by deriving a new "oracle maximal inequality" for a finite class of submartingales. This is achieved via integration by parts rather than a simplistic application of the triangle inequality. Consequently, we obtain a generalization of Lenglart's inequality for discrete-time martingales, extending it from the one-dimensional case to finite-dimensional settings, and further to certain infinite-dimensional cases through a "finite approximation device". The primary applications include several weak convergence theorems for sequences of separable martingale random fields under the uniform topology. In particular, new results are established for i.i.d. sequences, including a necessary and sufficient condition for a countable class of functions to possess the Donsker property. Additionally, we provide new moment bounds for the supremum of empirical processes indexed by classes of sets or functions.

From the oracle maximal inequality to martingale random fields via finite approximation from below

Abstract

A novel approach is proposed to establish a sharp upper bound on the expected supremum of a separable martingale random field, serving as an alternative to classical universal entropy-based methods. The proposed approach begins by deriving a new "oracle maximal inequality" for a finite class of submartingales. This is achieved via integration by parts rather than a simplistic application of the triangle inequality. Consequently, we obtain a generalization of Lenglart's inequality for discrete-time martingales, extending it from the one-dimensional case to finite-dimensional settings, and further to certain infinite-dimensional cases through a "finite approximation device". The primary applications include several weak convergence theorems for sequences of separable martingale random fields under the uniform topology. In particular, new results are established for i.i.d. sequences, including a necessary and sufficient condition for a countable class of functions to possess the Donsker property. Additionally, we provide new moment bounds for the supremum of empirical processes indexed by classes of sets or functions.

Paper Structure

This paper contains 20 sections, 14 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Motivation for this research
  3. An approach via finite approximation from below
  4. Announcements of an oracle maximal inequality and its consequence
  5. Some more preliminaries
  6. Announcements of results for i.i.d. random sequences
  7. A new central limit theorem for i.i.d. separable random fields
  8. A new Donsker theorem for empirical processes
  9. Supremal inequalities for empirical processes
  10. An extension of Jain-Marcus' theorem
  11. Organization of the paper
  12. Proof of the finite approximation device
  13. Proofs of the OMI and its consequence
  14. Infinite-dimensional martingales
  15. Supremal inequalities
  16. ...and 5 more sections

Key Result

Lemma 4

Let a constant $p\geq 1$ be given. Let $X=\{ X(\theta); \ \theta \in \Theta \}$ be a random field indexed by a non-empty set $\Theta$. (i) If $\Theta={\Bbb N}$ (that is, $X$ is actually a sequence, rather than a "general" countable set) of random variables, then it holds that allowing the possibility that the both sides are $\infty$. (ii) Suppose that the random field $X$ is separable with respec

Theorems & Definitions (34)

  • Definition 1: Separable random field
  • Definition 2
  • Remark 3
  • Lemma 4: Finite approximation device
  • Remark 5
  • Remark 6
  • Lemma 7: OMI for submartingales
  • Lemma 8: Maximal inequality for martingales
  • Theorem 9
  • Theorem 10
  • ...and 24 more