On convergence to stationarity of fractional Brownian storage

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.