On convergence to stationarity of fractional brownian storage

Michel Mandjes, Ilkka Norros, Peter Glynn

    Research output: Contribution to journalArticleScientificpeer-review

    11 Citations (Scopus)


    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.
    Original languageEnglish
    Pages (from-to)1385-1403
    Number of pages19
    JournalAnnals of Applied Probability
    Issue number4
    Publication statusPublished - 2009
    MoE publication typeA1 Journal article-refereed


    • Convergence to stationarity
    • Fractional brownian motion
    • Large deviations
    • Storage process


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