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2 results

  • Sparre Andersen identity and the last passage time

  • Part of
  • Jevgenijs Ivanovs
  • Journal:
    Journal of Applied Probability / Volume 53 / Issue 2 / June 2016
    Published online by Cambridge University Press:
    21 June 2016, pp. 600-605
    Print publication:
    June 2016

  • It is shown that the celebrated result of Sparre Andersen for random walks and Lévy processes has intriguing consequences when the last time of the process in (-∞, 0], say σ, is added to the picture. In the case of no positive jumps this leads to six random times, all of which have the same distribution—the uniform distribution on [0, σ]. Surprisingly, this result does not appear in the literature, even though it is based on some classical observations concerning exchangeable increments.

  • On the time spent above a level by Brownian motion with negative drift

  • J.-P. Imhof
  • Journal:
    Advances in Applied Probability / Volume 18 / Issue 4 / December 1986
    Published online by Cambridge University Press:
    01 July 2016, pp. 1017-1018
    Print publication:
    December 1986
    • Article
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  • Limit theorems of Berman involve the total time spent by Brownian motion with negative drift above a fixed or exponentially distributed negative level. We give explicitly the probability densities and distribution functions, obtained via an equivalence of laws.