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.
This is a preview of subscription content, access via your institution.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
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/s004400000077
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s004400000077
Mathematics Subject Classification (2000)
- Primary: 60J65, 60F65, 60J55
Key words and phrases
- Planar Brownian motion
- Additive functional
- Almost sure limit theorems
- Kallianpur-Robbins law
- Ratio ergodic theorem
- Chacon-Ornstein Theorem
- Log averaging methods
- Higher order logarithmic density