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Sample path behaviour in connection with generalized arcsine laws. (English) Zbl 0832.60049

Summary: Let \(G = (G(t), t \geq 0)\) 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 for \(G(t)/t\) to converge in law as \(t \to \infty\) 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 of \(G\). We obtain a similar result for \(A^+ = (A^+ (t), t \geq 0)\), the time spent in \([0, \infty)\) by a real-valued diffusion process, in connection with Watanabe’s recent extension of Lévy’s second arcsine law.

MSC:

60G17 Sample path properties
60J25 Continuous-time Markov processes on general state spaces
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