Last exit times for random walks

R. A. Doney

Stochastic Processes and their Applications, 1989, vol. 31, issue 2, 321-331

Abstract: If{Sn, n [greater-or-equal, slanted] 0} is a random walk which drifts to +[infinity], a last exit occurs at (n, Sn) if Sm > Sn for allm> n. In analogy with the more familiar first exits, it is shown that the set of all such (n, Sn) forms a modified two dimensional renewal process on [0, [infinity]) x (-[infinity], [infinity]). Furthermore the interpoint distribution is the same as in the first exit case, and the projection onto the time axis is stationary. The asymptotic behaviour of the distributions of the time at which the kth last exit occurs and the time at which the last exit from (-[infinity], a] occur are given (for fixed k and a respectively) whenever the left-hand tail of the step distribution is either regularly varying or obeys a Cramér-type condition.

Keywords: random; walk; exit; times; passage; times (search for similar items in EconPapers)
Date: 1989
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