### Abstract

With M(t):=sup

_{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 |
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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 |

### Keywords

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

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## Cite this

Mandjes, M., Norros, I., & Glynn, P. (2009). On convergence to stationarity of fractional brownian storage.

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