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Asymptotic behavior of the number of windings of the planar Brownian path. (Comportement asymptotique du nombre de tours effectués par la trajectoire brownienne plane.) (French) Zbl 0810.60077

Azéma, Jacques (ed.) et al., Séminaire de Probabilités XXVIII. Berlin: Springer-Verlag. Lect. Notes Math. 1583, 164-171 (1994).
The authors are interested in the almost sure asymptotic behaviour of the angular part \(\theta_ t\) of a planar Brownian motion \(Z\), \(\theta_ 0 = 0\). The precise statement is the following: let \(f\) be a positive increasing function, then \(\limsup_{t \to \infty} {\theta_ t \over f(t) \ln t} = 0\) or \(\infty\) iff \(\int^ \infty_ \bullet {dt \over tf(t) \ln t}\) is finite or infinite. The proof is based on some estimates established by Spitzer and properties of the Cauchy process. The authors deduce from the previous result the behaviour of \((\theta_ t; t \to 0)\), and also investigate the asymptotic property of the “clock” \(H_ t = \int^ t_ 0 | Z_ s |^{-2} ds\) when \(t \to \infty\): \(\limsup_{t \to \infty} {H_ t \over (f(t) \ln t)^ 2} = 0\) or \(\infty \Leftrightarrow \int^ \infty_ \bullet {dt \over tf(t) \ln t}\) is finite or infinite.
For the entire collection see [Zbl 0797.00020].
Reviewer: P.Vallois (Nancy)

MSC:

60J65 Brownian motion
60G17 Sample path properties
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