## 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):= sup_{0≤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(−θt^{2−2H} ), 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.
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