Abstract
Consider a Brownian motion W in C started from 0 and run for time 1. Let A(1), A(2), . . . denote the bounded connected components of C - W([0, 1]). Let R(i) (resp. r(i)) denote the out-radius (resp. in-radius) of A(i) for i ? N. Our main result is that E[ i R(i)2| log R(i)|?] < ∞ for any Θ < 1. We also prove that Σi r(i)2| log r(i)| = ∞ almost surely. These results have the interpretation that most of the components A(i) have a rather regular or round shape.
Original language | English (US) |
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Pages (from-to) | 882-908 |
Number of pages | 27 |
Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |
Volume | 55 |
Issue number | 2 |
DOIs | |
State | Published - May 1 2019 |
Keywords
- Complementary components of planar Brownian motion
- Planar Brownian motion
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty