Sample path behaviour in connection with generalized arcsine laws

Abstract

LetG=(G(t),t≧0) be the process of last passage times at some fixed point of a Markov process. The Dynkin-Lamperti theorem provides a necessary and sufficient condition forG(t)/t to converge in law ast→∞ to some non-degenerate limit (which is then a generalized arcsine law). Under this condition, we give a simple integral test that characterizes the lower-functions ofG. We obtain a similar result forA+=(A+(t),t≧0), the time spent in [0, ∞) by a real-valued diffusion process, in connection with Watanabe's recent extension of Levy's second arcsine law.