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Sample path properties of the Cauchy principal values of Brownian and random walk local times are studied. We establish LIL type results (without exact constants). Large and small increments are discussed. A strong approximation result between the above two processes is also proved.

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Summary This article provides a glimpse of some of the highlights of the joint work of Endre Csáki and Pál Révész since 1979.  The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.

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Tudor, C. A. and Xiao, Y. , Sample path properties of bifractional Brownian motion, Bernoulli , 13 (2007), 1023–1052. MR 2009e :60089 Xiao Y. Sample path properties of

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Tudor, C. A. and Xiao, Y. , Sample path properties of bifractional Brownian motion, Bernoulli , 13 (2007), no. 4, 1023–1052. MR 2364225 Xiao Y. Sample path

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, M., Földes, A. and Shi, Z., Path properties of Cauchy's principal values related to local time, Studia Sci. Math. Hungar. 38 (2001), 149-169. MR 2002j :60149 Path properties of Cauchy's principal values related to local

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properties of bifractional Brownian motion, Bernoulli , 13 (2007), 1023–1052. MR 2009e :60089 Xiao Y. Sample path properties of bifractional Brownian motion

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