On convergence to stationarity of fractional Brownian storage

TL;DR

This paper investigates how quickly the distribution of the maximum of a fractional Brownian motion with negative drift converges to its stationary distribution, revealing an exponential decay rate linked to busy period asymptotics in fBm-driven queues.

Contribution

It establishes the decay rate of convergence metrics for fractional Brownian storage, connecting it to busy period tail asymptotics and extending the relation to other Gaussian process settings.

Findings

Convergence metrics decay roughly as exp(-θ t^{2-2H})

Decay rate θ matches the tail distribution of busy periods in fBm queues

Relation holds under G"artner--Ellis-type conditions in broader settings

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 G\"artner--Ellis-type conditions are fulfilled.