Sample path properties of reflected Gaussian processes

TL;DR

This paper investigates the long-term behavior of a stationary Gaussian queueing process, providing criteria for exceedance probabilities and establishing a law of the iterated logarithm for last passage times, extending known results beyond fractional Brownian motion.

Contribution

It introduces new criteria for exceedance probabilities and proves a law of the iterated logarithm for Gaussian processes, generalizing previous results to broader Gaussian models.

Findings

Criteria for zero-one laws of exceedance probabilities.

An Erd"os-Rényi type law of the iterated logarithm for last passage times.

Extension of results from fractional Brownian motion to general Gaussian processes.

Abstract

We consider a stationary queueing process $Q_X$ fed by a centered Gaussian process $X$ with stationary increments and variance function satisfying classical regularity conditions. A criterion when, for a given function $f$, $\mathbb P (Q_{X}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1 is provided. Furthermore, an Erd\"os-R\'ev\'esz type law of the iterated logarithm is proven for the last passage time $\xi (t) = \sup\{s:0\le s\le t, Q_{X}(s)\ge f(s)\}$. Both of these findings extend previously known results that were only available for the case when $X$ is a fractional Brownian motion.