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An exponential inequality for orthomartingale differences random fields and some applications

Davide Giraudo

Abstract

In this paper, we establish an exponential inequality for random fields, which is applied in the context of convergence rates in the law of large numbers and Hölderian weak invariance principle.

An exponential inequality for orthomartingale differences random fields and some applications

Abstract

In this paper, we establish an exponential inequality for random fields, which is applied in the context of convergence rates in the law of large numbers and Hölderian weak invariance principle.

Paper Structure

This paper contains 11 sections, 9 theorems, 51 equations.

Table of Contents

  1. Statement of the exponential inequality
  2. Goal and motivations
  3. Exponential inequality for orthomartingales
  4. Application to limit theorems
  5. Convergence rates in the law of large numbers
  6. Application to Hölderian invariance principle
  7. Proofs
  8. Proof of Theorem \ref{['thm:deviation_inequality_orthomartingale']}
  9. Proof of Theorem \ref{['thm:large_deviation']}
  10. Proof of Proposition \ref{['prop:tightness_partial_sums']}
  11. Proof of Theorem \ref{['thm:WIP_Holder']}

Key Result

Theorem 1.6

Let $\left(X_{\bm{i}}\right)_{\bm{i}\in\mathbb Z^d}$ be an orthomartingale differences random field such that for all $\bm{n}\in\mathbb N^d$ and all $\bm{k}\in\mathbb Z^d$, Then the following inequality holds for all $x,y>0$: where $A_d$, $B_d$ and $C_d$ depend only on $d$, $p_d=2d$ and $\left|\bm{n}\right|=\prod_{q=1}^dn_q$.

Theorems & Definitions (20)

  • Definition 1.1
  • Definition 1.2
  • Example 1.3
  • Example 1.4
  • Definition 1.5
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 2.1
  • ...and 10 more