On convergence to stationarity of fractional Brownian storage

With M(t) := sups2[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(−#t2−2H), where # is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [16]. 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.