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1 results

  • On the arc-sine laws for Lévy processes

  • R. K. Getoor, M. J. Sharpe
  • Journal:
    Journal of Applied Probability / Volume 31 / Issue 1 / March 1994
    Published online by Cambridge University Press:
    14 July 2016, pp. 76-89
    Print publication:
    March 1994

  • Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.