The Annals of Applied Probability

On convergence to stationarity of fractional Brownian storage

Michel Mandjes, Ilkka Norros, and Peter Glynn

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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(−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.

Article information

Ann. Appl. Probab., Volume 19, Number 4 (2009), 1385-1403.

First available in Project Euclid: 27 July 2009

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Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G18: Self-similar processes 90B05: Inventory, storage, reservoirs

Convergence to stationarity fractional Brownian motion storage process large deviations


Mandjes, Michel; Norros, Ilkka; Glynn, Peter. On convergence to stationarity of fractional Brownian storage. Ann. Appl. Probab. 19 (2009), no. 4, 1385--1403. doi:10.1214/08-AAP578.

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