Stable windings at the origin |
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Authors: | Andreas E. Kyprianou Stavros M. Vakeroudis |
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Affiliation: | 1. Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY, UK;2. Department of Mathematics, Track: Statistics and Actuarial-Financial Mathematics, University of the Aegean, 83200 Karlovasi, Samos, Greece |
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Abstract: | In 1996, Bertoin and Werner demonstrated a functional limit theorem, characterising the windings of planar isotropic stable processes around the origin for large times, thereby complementing known results for planar Brownian motion. The question of windings at small times can be handled using scaling. Nonetheless we examine the case of windings at the origin using new techniques from the theory of self-similar Markov processes. This allows us to understand upcrossings of (not necessarily symmetric) stable processes over the origin for large and small times in the one-dimensional setting. |
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Keywords: | primary 60J30 60G18 secondary 60J15 Stable processes Winding numbers Self-similarity Lamperti transform Duality Time change Riesz–Bogdan–?ak transform Upcrossings |
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