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Moderate deviations of suprema of Gaussian processes A cyclic approximation criterion

Michel Weber

TL;DR

This work develops a comprehensive framework for moderate deviations of suprema of Gaussian processes, covering trigonometrical and almost periodic settings. It combines decoupling inequalities with a novel approximation criterion that replaces almost periodic polynomials by cyclic stationary surrogates via modulable Diophantine approximation, yielding explicit exponential-error bounds. A central contribution is the semi.asymp.bound.cor result, which bridges almost periodic and cyclic models, augmented by a detailed analysis of decoupling coefficients and Gaussian semi-norms. The paper also advances lattice-analytic methods by establishing localized Kronecker-type theorems and correlation controls for almost periodic Gaussian polynomials with linearly independent frequencies, enabling quantitative bounds on supremum probabilities in a broad class of models.

Abstract

We study moderate deviations of suprema of parametrized sequences of sample bounded Gaussian processes $\{X _x(t), t\in T _x\}$, and first present recent sharp bounds in simple cases. In the almost periodic case, we prove an approximation theorem. We introduce a modulable diophantine approximation. Finally we study for general non-vanishing coefficient sequences, the behavior along lattices of almost periodic Gaussian polynomials with linearly independent frequencies, and use a lattice localized version of Kronecker's theorem.

Moderate deviations of suprema of Gaussian processes A cyclic approximation criterion

TL;DR

This work develops a comprehensive framework for moderate deviations of suprema of Gaussian processes, covering trigonometrical and almost periodic settings. It combines decoupling inequalities with a novel approximation criterion that replaces almost periodic polynomials by cyclic stationary surrogates via modulable Diophantine approximation, yielding explicit exponential-error bounds. A central contribution is the semi.asymp.bound.cor result, which bridges almost periodic and cyclic models, augmented by a detailed analysis of decoupling coefficients and Gaussian semi-norms. The paper also advances lattice-analytic methods by establishing localized Kronecker-type theorems and correlation controls for almost periodic Gaussian polynomials with linearly independent frequencies, enabling quantitative bounds on supremum probabilities in a broad class of models.

Abstract

We study moderate deviations of suprema of parametrized sequences of sample bounded Gaussian processes , and first present recent sharp bounds in simple cases. In the almost periodic case, we prove an approximation theorem. We introduce a modulable diophantine approximation. Finally we study for general non-vanishing coefficient sequences, the behavior along lattices of almost periodic Gaussian polynomials with linearly independent frequencies, and use a lattice localized version of Kronecker's theorem.

Paper Structure

This paper contains 16 sections, 21 theorems, 196 equations.

Table of Contents

  1. Introduction-Results
  2. The trigonometrical case.
  3. The almost periodic case.
  4. An approximation criterion.
  5. Decoupling inequalities of Gaussian processes.
  6. Proof of Theorem \ref{['semi.asymp.bound.cor']}
  7. Intermediate results
  8. Proof of Theorem \ref{['semi.asymp.bound.cor']}
  9. Proof of Theorem \ref{['sup.kappa.th']}.
  10. Controlling approximation
  11. An intermediate approximation result.
  12. Gauss-ian semi-norms
  13. Almost periodic Gaussian polynomials with linearly independent frequencies along lattices
  14. Localized versions of Kronecker's theorem.
  15. Proofs of Theorems \ref{['t3']} and \ref{['ak.cos']}
  16. ...and 1 more sections

Key Result

Theorem 1.1

(i) For any positive real $\Theta$, and any $n\ge 2$, (ii) Set $b_n= ( \log \frac{n^2}{4\pi \log n} )^{1/2}$. Given any positive real ${\varepsilon}$, for $n$ sufficiently large, $n\ge n({\varepsilon})$, and all $x$ such that $x\ge -b_n^2$,

Theorems & Definitions (27)

  • Theorem 1.1: Weber W1a, Th. 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4: S, Lemma 1, p. 468
  • Theorem 1.5: W, Th. 4.1
  • Theorem 1.6
  • Corollary 1.8
  • Theorem 1.9: Approximation criterion
  • Theorem 2.1: KLS, Th. 3
  • Theorem 2.2: KLS, Th. 1
  • ...and 17 more