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(−θt), 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.