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A Third Arc‐Sine Theorem
Author(s) -
Doney R. A.,
Marchal P.
Publication year - 2003
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609303002157
Subject(s) - mathematics , sine , brownian motion , arc (geometry) , random walk , inverse trigonometric functions , zero (linguistics) , mathematics subject classification , conditional expectation , mathematical analysis , pure mathematics , statistics , geometry , linguistics , philosophy
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. 2000 Mathematics Subject Classification 60G50.