On convergence to stationarity of fractional Brownian storage

Abstract

With $M(t):={\sup }_{s\in [0,t]}A(s)-s$ denoting the running maximum of a fractional Brownian motion $A(\cdot )$ with negative drift, this paper studies the rate of convergence of $\mathbb{P}(M(t)>x)$ to $\mathbb{P}(M>x)$. We define two metrics that measure the distance between the (complementary) distribution functions $\mathbb{P}(M(t)>\cdot )$ and $\mathbb{P}(M>\cdot )$. Our main result states that both metrics roughly decay as $\exp (-\vartheta t^{2-2H})$, where $\vartheta$ is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [Stochastic Process. Appl. (2006) 116 1269--1293]. The proofs extensively rely on application of the well-known large deviations theorem for Gaussian processes. We also show that the identified relation between the decay of the convergence metrics and busy-period asymptotics holds in other settings as well, most notably when Gärtner--Ellis-type conditions are fulfilled.

Published in at http://dx.doi.org/10.1214/08-AAP578 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (1)

• Sample-path large deviations for tandem and priority queues with Gaussian inputs