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Large Deviations for High Minima of Gaussian Processes with Nonnegatively Correlated Increments

Zachary Selk

Abstract

In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such processes on intervals $[a,b]$ where $b-a$ is either less than the increment or twice the increment, assuming stationarity of the increments. As a chief example, we consider fractional Brownian motion and fractional Gaussian noise for $H\geq 1/2$.

Large Deviations for High Minima of Gaussian Processes with Nonnegatively Correlated Increments

Abstract

In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such processes on intervals where is either less than the increment or twice the increment, assuming stationarity of the increments. As a chief example, we consider fractional Brownian motion and fractional Gaussian noise for .

Paper Structure

This paper contains 6 sections, 8 theorems, 50 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Results
  3. Processes with Nonnegatively Correlated Increments
  4. Increment of Processes with Nonnegatively Correlated Increments
  5. Fractional Brownian Motion and Fractional Gaussian Noise
  6. Further Questions

Key Result

Theorem 2.1

(Theorem 4.3 Gennady-High-Level-Sets) Let $X(t)$ be a continuous centered Gaussian process on $[a,b]$ with covariance function $R(s,t)$. Then $\mu^\ast$ is an optimal measure in problem $\sigma_\ast^2(a,b)$ defined in equation eq:optimization if and only if for $\mu^\ast$-a.e. $t\in [a,b]$, where

Figures (6)

  • Figure 1: The increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$
  • Figure 2: The first derivative of the increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$
  • Figure 3: The second derivative of the increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$
  • Figure 4: The covariance function $\Gamma$ for fBm with $2H=1.5$, $h=1$
  • Figure 5: The function $\gamma$ for fBm with $2H=1.5$, $h=1$
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 10 more