### Abstract

For families of random walks {S(a)k } with ES(a) k = -ka <0 we consider their maxima M(a) = supk = 0 S(a)k .
We investigate the asymptotic behaviour of M(a) as a ¿ 0 for asymptotically stable random walks. This problem appeared first in the 1960’s in the analysis of a single-server queue when the traffic load tends to 1 and since then is referred to as the heavy-traffic
approximation problem. Kingman and Prokhorov suggested two different approaches
which were later followed by many authors. We give two elementary proofs of our main result, using each of these approaches. It turns out that the main technical difficulties in both proofs are rather similar and may be resolved via a generalisation of the Kolmogorov inequality to the case of an infinite variance. Such a generalisation is also obtained in this note

Original language | English |
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Place of Publication | Eindhoven |

Publisher | Eurandom |

Number of pages | 9 |

Publication status | Published - 2009 |

### Publication series

Name | Report Eurandom |
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Volume | 2009005 |

ISSN (Print) | 1389-2355 |

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## Cite this

Shneer, V., & Wachtel, V. (2009).

*Heavy-traffic analysis of the maximum of an asymptotically stable random walk*. (Report Eurandom; Vol. 2009005). Eurandom.