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Mon, 03 Apr 2000 00:00:00 +0200Mon, 03 Apr 2000 00:00:00 +0200Symmetry properties of average densities and tangent measure distributions of measures on the line
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/818
Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/818Mon, 03 Apr 2000 00:00:00 +0200Tangent measure distributions of fractal measures
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/821
Tangent measure distributions are a natural tool to describe the local geometry of arbitrary measures of any dimension. We show that for every measure on a Euclidean space and every s, at almost every point, all s-dimensional tangent measure distributions define statistically self-similar random measures. Consequently, the local geometry of general measures is not different from the local geometry of self-similar sets. We illustrate the strength of this result by showing how it can be used to improve recently proved relations between ordinary and average densities.Peter Mörters; David Preisspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/821Mon, 03 Apr 2000 00:00:00 +0200Pathwise Kallianpur-Robbins laws for Brownian motion in the plane
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/840
The Kallianpur-Robbins law describes the long term asymptotic behaviour of the distribution of the occupation measure of a Brownian motion in the plane. In this paper we show that this behaviour can be seen at every typical Brownian path by choosing either a random time or a random scale according to the logarithmic laws of order three. We also prove a ratio ergodic theorem for small scales outside an exceptional set of vanishing logarithmic density of order three.Peter Mörterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/840Fri, 28 Jan 2000 00:00:00 +0100