Upper tail probabilities of integrated Brownian motions

TL;DR

This paper derives new upper tail probability bounds for integrated Brownian motions in different norms, linking small ball and large ball probabilities, with applications to eigenvalues and Laplace transforms.

Contribution

It introduces a novel method connecting small ball and upper tail probabilities for Gaussian processes, providing explicit bounds and asymptotic behaviors.

Findings

Explicit bounds for the largest eigenvalue of the covariance operator

Connections established between small and large ball probabilities

Asymptotic behaviors of Laplace transforms for integrated Brownian motions

Abstract

We obtain new upper tail probabilities of $m$-times integrated Brownian motions under the uniform norm and the $L^p$ norm. For the uniform norm, Talagrand's approach is used, while for the $L^p$ norm, Zolotare's approach together with suitable metric entropy and the associated small ball probabilities are used. This proposed method leads to an interesting and concrete connection between small ball probabilities and upper tail probabilities (large ball probabilities) for general Gaussian random variable in Banach spaces. As applications, explicit bounds are given for the largest eigenvalue of the covariance operator, and appropriate limiting behaviors of the Laplace transforms of $m$-times integrated Brownian motions are presented as well.