### Abstract

_{s∈[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(−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.

Original language | English |
---|---|

Pages (from-to) | 1385-1403 |

Number of pages | 19 |

Journal | Annals of Applied Probability |

Volume | 19 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2009 |

MoE publication type | A1 Journal article-refereed |

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### Keywords

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

### Cite this

*Annals of Applied Probability*,

*19*(4), 1385-1403. https://doi.org/10.1214/08-AAP578

}

*Annals of Applied Probability*, vol. 19, no. 4, pp. 1385-1403. https://doi.org/10.1214/08-AAP578

**On convergence to stationarity of fractional brownian storage.** / Mandjes, Michel; Norros, Ilkka; Glynn, Peter.

Research output: Contribution to journal › Article › Scientific › peer-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 -