Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

Details Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations Veraverbeke's theorem at large - On the maximum of some processes with negative drift and heavy tail innovations

2008-02-25

arxiv.org/abs/0802.3638v1

Mathematics

Probability

86 pages, 4 figures

Scientific paper

Veraverbeke's (1977) theorem relates the tail of the distribution of the supremum of a random walk with negative drift to the tail of the distribution of its increments, or equivalently, the probability that a centered random walk with heavy-tail increments hits a moving linear boundary. We study similar problems for more general processes. In particular, we derive an analogue of Veraverbeke's theorem for fractional integrated ARMA models without prehistoric influence, when the innovations have regularly varying tails. Furthermore, we prove some limit theorems for the trajectory of the process, conditionally on a large maximum. Those results are obtained by using a general scheme of proof which we present in some detail and should be of value in other related problems.

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