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On the length of the homotopic Brownian word in the thrice punctured sphere
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  • Published: August 1998

On the length of the homotopic Brownian word in the thrice punctured sphere

  • Jean-Claude Gruet1 

Probability Theory and Related Fields volume 111, pages 489–516 (1998)Cite this article

Abstract.

We consider the word associated to the homotopic class of the Brownian path (properly closed) in the thrice punctured sphere. We prove that its length has almost surely the same behaviour as a totally asymmetric Cauchy process on the line. More precisely, the liminf has the same normalization in t log(t) and the limsup can be described by the same integral test. They are the Brownian motion counterparts of some Lévy and Khintchine results on continued fraction expansions.

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  1.  Laboratoire de Probabilités, Université Pierre et Marie Curie, 4, Place Jussieu, F-75252 Paris, France, e-mail: gruet@ccr.jussieu.fr, , , , , , FR

    Jean-Claude Gruet

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  1. Jean-Claude Gruet
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Received: 17 December 1996 / Revised version: 23 February 1998

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Gruet, JC. On the length of the homotopic Brownian word in the thrice punctured sphere. Probab Theory Relat Fields 111, 489–516 (1998). https://doi.org/10.1007/s004400050175

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  • Issue Date: August 1998

  • DOI: https://doi.org/10.1007/s004400050175

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  • Mathematics Subject Classification (1991): 60J65 60E07 58G32 11A55
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