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A THIRD ARC-SINE THEOREM

Published online by Cambridge University Press:  09 June 2003

R. A. DONEY  and
P. MARCHAL
R. A. DONEY
Affiliation:
Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL [email protected]
P. MARCHAL
Affiliation:
CNRS, DMA, École normale supérieure, 45 rue d'Ulm, 75230 Paris cedex 05, [email protected]
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Abstract

As well as the arc-sine laws for the time spent positive before time 1 and the time at which the maximum over [0, 1] occurs, Paul Lévy also established a third arc-sine law for Brownian motion; this relates to the time of the last visit to zero before time 1. For random walks, analogues of the first two of the these results are well known. Here it is shown that for any random walk satisfying Spitzer's condition there is an analogue of the third, but care has to be taken in its formulation. An associated conditional uniform law and some applications to stable processes are also given.

Keywords

60G50
Type
Notes and Papers
Information
Bulletin of the London Mathematical Society , Volume 35 , Issue 4 , July 2003 , pp. 536 - 540
Copyright
© The London Mathematical Society 2003

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