Probability Theory and Related Fields

, Volume 106, Issue 1, pp 1–38

Two renewal theorems for general random walks tending to infinity

  • Harry Kesten
  • R. A. Maller

DOI: 10.1007/s004400050056

Cite this article as:
Kesten, H. & Maller, R. Probab Theory Relat Fields (1996) 106: 1. doi:10.1007/s004400050056


Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk Sn into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that Sn→∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of Sn, are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite.

A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given.

Mathematics Subject classification (1991): 60K0560J1560F1560G4060G50

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Harry Kesten
    • 1
  • R. A. Maller
    • 1
  1. 1.Department of Mathematics, Cornell University, Ithaca, NY 14853-7901, USA, (e-mail: kesten@ AU