On convergence to stationarity of fractional brownian storage

Michel Mandjes, Ilkka Norros, Peter Glynn

Research output: Contribution to journalArticleScientificpeer-review

7 Citations (Scopus)

Abstract

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
Volume19
Issue number4
DOIs
Publication statusPublished - 2009
MoE publication typeA1 Journal article-refereed

Fingerprint

Stationarity
Busy Period
Fractional
Metric
Complementary function
Decay
Fractional Brownian Motion
Gaussian Process
Large Deviations
Decay Rate
Queue
Stochastic Processes
Tail
Distribution Function
Rate of Convergence
Theorem

Keywords

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

Cite this

Mandjes, Michel ; Norros, Ilkka ; Glynn, Peter. / On convergence to stationarity of fractional brownian storage. In: Annals of Applied Probability. 2009 ; Vol. 19, No. 4. pp. 1385-1403.
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On convergence to stationarity of fractional brownian storage. / Mandjes, Michel; Norros, Ilkka; Glynn, Peter.

In: Annals of Applied Probability, Vol. 19, No. 4, 2009, p. 1385-1403.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - On convergence to stationarity of fractional brownian storage

AU - Mandjes, Michel

AU - Norros, Ilkka

AU - Glynn, Peter

PY - 2009

Y1 - 2009

N2 - 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.

AB - 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.

KW - Convergence to stationarity

KW - Fractional brownian motion

KW - Large deviations

KW - Storage process

U2 - 10.1214/08-AAP578

DO - 10.1214/08-AAP578

M3 - Article

VL - 19

SP - 1385

EP - 1403

JO - Annals of Applied Probability

JF - Annals of Applied Probability

SN - 1050-5164

IS - 4

ER -