A Joint Integral Test for the Locations of Extrema for Brownian Motion |
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Authors: | Youssef Randjiou |
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Affiliation: | (1) Laboratoire de Probabilités, Université Paris VI, 4 Place Jussieu, F–75252 Paris Cedex 05, France |
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Abstract: | Let μ+(t) and μ−(t) be the locations of the maximum and minimum, respectively, of a standard Brownian motion in the interval [0,t]. We establish a joint integral test for the lower functions of μ+(t) and μ−(t), in the sense of Paul Lévy. In particular, it yields the law of the iterated logarithm for max(μ+(t),μ−(t)) as a straightforward consequence. Our result is in agreement with well-known theorems of Chung and Erdős [(1952) Trans. Amer. Math. Soc. 72, 179–186.], and Csáki, F?ldes and Révész [(1987) Prob. Theory Relat. Fields 76, 477–497]. |
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Keywords: | Brownian motion Lévy’ s lower class joint integral test law of the iterated logarithm |
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