Local independence of fractional Brownian motion |
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Authors: | Ilkka Norros Eero Saksman |
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Affiliation: | 1. VTT Technical Research Centre of Finland, P.O. Box 1000, 02044 VTT, Finland;2. University of Helsinki, Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hällströmin katu 2b), FIN-00014 University of Helsinki, Finland |
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Abstract: | Let σ(t,t′) be the sigma-algebra generated by the differences Xs−Xs′ with s,s′∈(t,t′), where (Xt)−∞<t<∞ is the fractional Brownian motion with Hurst index H∈(0,1). We prove that for any two distinct timepoints t1 and t2 the sigma-algebras σ(t1−ε,t1+ε) and σ(t2−ε,t2+ε) are asymptotically independent as ε↘0. We show the independence in the strong sense that Shannon’s mutual information between the two σ-algebras tends to zero as ε↘0. Some generalizations and quantitative estimates are also provided. |
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Keywords: | 60G15 (60G18 94A99 60H99) |
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