Article

Extremes

, Volume 14, Issue 1, pp 63-125

Veraverbeke’s theorem at large: on the maximum of some processes with negative drift and heavy tail innovations

  • Ph. BarbeAffiliated withCNRS (UMR 8088)
  • , W. P. McCormickAffiliated withDept. of Statistics, University of Georgia Email author 

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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.

Keywords

Maximum of random walk Heavy tail Fractional ARIMA process Long range dependence Boundary crossing probability Nonlinear renewal theory

AMS 2000 Subject Classifications

Primary—60G50; Secondary—60F99 60G99 60K30 62P05 62M10 26A12 26A33