On convergence to stationarity of fractional Brownian storage

With M(t):=sups∈[0, t]A(s)−s denoting the running maximum of a fractional Brownian motion A(⋅) with negative drift, this paper studies the rate of convergence of P(M(t)>x) to P(M>x). We define two metrics that measure the distance between the (complementary) distribution functions P(M(t)>⋅) and P(M>⋅). Our main result states that both metrics roughly decay as exp(−ϑvt2−2H), where ϑ 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.