A Joint Integral Test for the Locations of Extrema for Brownian Motion

Youssef Randjiou ()
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Youssef Randjiou: Université Paris VI

Journal of Theoretical Probability, 2006, vol. 19, issue 3, 701-720

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, Földes and Révész [(1987) Prob. Theory Relat. Fields 76, 477–497].

Keywords: Brownian motion; Lévy’s lower class; joint integral test; law of the iterated logarithm; 60F15; 60J65 (search for similar items in EconPapers)
Date: 2006
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DOI: 10.1007/s10959-006-0030-5

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