The Annals of Probability

Further Asymptotic Laws of Planar Brownian Motion

Jim Pitman and Marc Yor

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The asymptotic distributions for large times of a variety of additive functionals of planar Brownian motion $Z$ are derived. Associated with each point in the plane, and with the point infinity, there is a complex Brownian motion governing the asymptotic behavior of windings of $Z$ close to that point. An independent Gaussian field over the plane governs fluctuations in local occupation times of $Z$, while a further independent family of complex Brownian sheets governs finer features of the windings of $Z$. These results unify and extend earlier results of Kallianpur and Robbins, Spitzer, Kasahara and Kotani, Messulam and the authors.

Article information

Ann. Probab., Volume 17, Number 3 (1989), 965-1011.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G65
Secondary: 60G55: Point processes 60F05: Central limit and other weak theorems 60H05: Stochastic integrals 60G44: Martingales with continuous parameter 60F17: Functional limit theorems; invariance principles

Winding numbers asymptotic distributions continuous martingales additive functionals Brownian motion singular integrals Brownian sheets


Pitman, Jim; Yor, Marc. Further Asymptotic Laws of Planar Brownian Motion. Ann. Probab. 17 (1989), no. 3, 965--1011. doi:10.1214/aop/1176991253.

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