ON THE ARC-SINE LAWS FOR LEVY PROCESSES

Let X be a Levy process on the real line, and let F(c) denote the generalized arc-sine law on [0, 1] with parameter c. Then t-1 integral-t/0 P0(X(s) > 0)ds --> c as t --> infinity is a necessary and sufficient condition for t-1 integral-t/0 1{Xs > 0} ds to converge in P0 law to F(c). Moreover, P0(X(t) > 0) = c for all t > 0 is a necessary and sufficient condition for t-1 integral-t/0 1{Xs > 0}ds under p0 to have law F(c) 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 Levy process version.