With M(t) := sups2[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(−#t2−2H), where # is the decay rate corresponding to the tail distribution of the busy period in an fBm-driven queue, which was computed recently [16]. 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 G¨artner-Ellis-type conditions are fulfilled.
Additional Metadata
Keywords fractional Brownian motion, convergence to stationarity, large deviations
MSC Gaussian processes (msc 60G15), Large deviations (msc 60F10), Queueing theory (msc 60K25)
THEME Logistics (theme 3), Energy (theme 4)
Publisher CWI
Series CWI. Probability, Networks and Algorithms [PNA]
Note Part of this work was done while MM was at Stanford University, Stanford, CA 94305, US.
Glynn, P, Mandjes, M.R.H, & Norros, I. (2008). On convergence to stationarity of fractional Brownian storage. CWI. Probability, Networks and Algorithms [PNA]. CWI.