Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations |
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Authors: | Ph Barbe W P McCormick |
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Institution: | 1.CNRS (UMR 8088),Paris,France;2.Dept. of Statistics,University of Georgia,Athens,USA |
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Abstract: | Veraverbeke’s (Stoch Proc Appl 5:27–37, 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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