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

TL;DR

This paper establishes large deviations principles for the high minima of Gaussian processes with nonnegative correlated increments, including fractional Brownian motion, on arbitrary intervals.

Contribution

It extends large deviations theory to Gaussian processes with nonnegative correlated increments and provides results for fractional Brownian motion and fractional Gaussian noise.

Findings

Large deviations principles are proved for high minima of such processes.

Results include increments on intervals with length less than or equal to the process's increment.

Applications to fractional Brownian motion and fractional Gaussian noise for H ≥ 1/2.

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 $H\geq 1/2$.