On the relative lengths of excursions derived from a stable subordinator

On the relative lengths of excursions derived from a stable subordinator

Report Number
469
Authors
Jim Pitman and Marc Yor
Citation
S{\'e}minaire de Probabilit{\'e}s XXXI, 287-305, Lecture Notes in Math. 1655, Springer, 1997
Abstract

Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times $T$ the distribution of relative excursion lengths prior to $T$ is the same as if $T$ were a fixed time. It follows that the generalized arc-sine laws of Lamperti extend to such random times $T$. For some other random times $T$, absolute continuity relations are obtained which relate the law of the relative lengths at time $T$ to the law at a fixed time.

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