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Almost sure Kallianpur–Robbins laws for Brownian motion in the plane
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  • Published: 12 February 2014

Almost sure Kallianpur–Robbins laws for Brownian motion in the plane

  • Peter Mörters1 

Probability Theory and Related Fields volume 118, pages 49–64 (2000)Cite this article

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Abstract

The Kallianpur–Robbins law describes the long term asymptotic behaviour of integrable additive functionals of Brownian motion in the plane. In this paper we prove an almost sure version of this result. It turns out that, differently from many known results, this requires an iterated logarithmic average. A similar result is obtained for the small scales asymptotic by means of an ergodic theorem of Chacon–Ornstein type, which allows an exceptional set of scales.

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Authors and Affiliations

  1. Universität Kaiserslautern, Fachbereich Mathematik, Postfach 3049, 67663 Kaiserslautern, Germany. e-mail: peter@mathematik.uni-kl.de, Germany

    Peter Mörters

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  1. Peter Mörters
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Received: 8 May 1998 / Revised version: 1 December 1999 / Published online: 8 August 2000

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Cite this article

Mörters, P. Almost sure Kallianpur–Robbins laws for Brownian motion in the plane. Probab Theory Relat Fields 118, 49–64 (2000). https://doi.org/10.1007/PL00008742

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  • Published: 12 February 2014

  • Issue Date: September 2000

  • DOI: https://doi.org/10.1007/PL00008742

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Keywords

  • Brownian Motion
  • Limit Theorem
  • Central Limit Theorem
  • Ergodic Theorem
  • Harmonic Measure
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