On Convergence to Stationarity of Fractional Brownian Storage

M. Mandjes, I. Norros, and P. W. Glynn

Annals of Applied Probability, Vol. 19, No. 4, 1385-1403 (2009)

With M(t):= sup0≤s≤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(−θt2−2H ), 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.