On the distribution of the last exit time over a slowly growing linear boundary for a Gaussian process

TL;DR

This paper establishes a limit theorem for the distribution of the last exit time of certain Gaussian stationary processes over a slowly increasing linear boundary, showing convergence to a Gumbel distribution.

Contribution

It provides the first limit theorem describing the asymptotic distribution of the last exit time over a slowly growing boundary for Gaussian processes.

Findings

Last exit time distribution converges to a Gumbel distribution.

Applicable to a class of Gaussian stationary processes.

Enhances understanding of boundary crossing behavior in Gaussian processes.

Abstract

For a class of Gaussian stationary processes, we prove a limit theorem on the convergence of the distributions of the scaled last exit time over a slowly growing linear boundary. The limit is a double exponential (Gumbel) distribution.