ON CONVERGENCE TO STATIONARITY OF FRACTIONAL BROWNIAN STORAGE

With M(t) := sup(s is an element of[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(-vt(2-2H)), where v is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [23] 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 Gartner-Ellis-type conditions are fulfilled.