On the arc-sine laws for Lévy processes
1994 ◽
Vol 31
(01)
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pp. 76-89
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Keyword(s):
The Real
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Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds under P 0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.
2013 ◽
Vol 444-445
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pp. 625-627
1966 ◽
Vol 62
(4)
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pp. 673-677
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1984 ◽
Vol 96
(2)
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pp. 213-222
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