Abstract
A celebrated theorem of Spitzer suggests that the number of windings made by a planar Brownian motion Z around the origin and taken in the logarithmic time-scale, is asymptotically close to a Cauchy process. The purpose of this paper is to show that this informal consideration can be made precise by introducing the Ornstein-Uhlenbeck process X(t)=e −t/2Z(et). This yields short proofs of known results as well as some new features on the asymptotic behaviour of the winding number (in distribution and pathwise).
Key Words
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Bertoin and W. Werner, ‘Comportement asymptotique du nombre de tours effectués par la trajectoire brownienne plane', Séminaire de Probabilités XXVIII, Lect. Notes in Math., Springer (1994).
L. Breiman, ‘A delicate law of the iterated logarithm for non-decreasing stable processes', Ann. Math. Statist. 39 (1968) 1818–1824. [correction id. 41 (1970) 1126–1127].
K.L. Chung, ‘On the maximum partial sums of sequences of independent random variables', Trans. Amer. Math. Soc. 64 (1948) 205–233.
R. Durrett, ‘A new proof of Spitzer's result on the winding of two-dimensional Brownian motion', Ann. Probab. 10 (1982) 244–246.
R. Durrett, ‘Brownian motion and martingales in analysis', Wadsworth (1984).
W.E. Feller, ‘A limit theorem for random variables with infinite moments', Amer. J. Math. 68 (1946) 257–262.
J. Franchi, ‘Théorème des résidus stochastique et asymptotique pour 1-formes sur S 2', Prépublication du Laboratoire de Probabilités 16, Université de Paris VI (1989).
B.E. Fristedt, 'sample functions of stochastic processes with stationary independent increments', Advances in Probability III, P. Ney, S. Port (eds) 241–396, Dekker (1974).
J.C. Gruet, ‘Enroulement du mouvement brownien plan autour de deux points', Thèse de Doctorat, Université de Paris VI (1990).
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1994 Springer-Verlag
About this paper
Cite this paper
Bertoin, J., Werner, W. (1994). Asymptotic windings of planar Brownian motion revisited via the Ornstein-Uhlenbeck process. In: Azéma, J., Yor, M., Meyer, P.A. (eds) Séminaire de Probabilités XXVIII. Lecture Notes in Mathematics, vol 1583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0073842
Download citation
DOI: https://doi.org/10.1007/BFb0073842
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-58331-8
Online ISBN: 978-3-540-48656-5
eBook Packages: Springer Book Archive