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Iterating Brownian Motions, Ad Libitum
Ecole Normale Super, Dept Math & Applicat, F-75230 Paris 05, France..
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory. Uppsala Univ, Dept Math, S-75106 Uppsala, Sweden..
2014 (English)In: Journal of theoretical probability, ISSN 0894-9840, E-ISSN 1572-9230, Vol. 27, no 2, 433-448 p.Article in journal (Refereed) Published
Abstract [en]

Let B-1, B-2, aEuro broken vertical bar be independent one-dimensional Brownian motions parameterized by the whole real line such that B (i) (0)=0 for every ia parts per thousand yen1. We consider the nth iterated Brownian motion W (n) (t)=B (n) (B (n-1)(a <-(B (2)(B (1)(t)))a <-)). Although the sequence of processes (W (n) ) (na parts per thousand yen1) does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W (n) converge to a random probability measure mu (a). We then prove that mu (a) almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W (a) of independent Brownian motions. We also prove that the collection of random variables (W (a)(t),taa"ea-{0}) is exchangeable with directing measure mu(infinity).

Place, publisher, year, edition, pages
2014. Vol. 27, no 2, 433-448 p.
Keyword [en]
Brownian motion, Iterated Brownian motion, Harris chain, Random measure, Exchangeability, Weak convergence, Local time, de Finetti-Hewitt-Savage theorem
National Category
Probability Theory and Statistics
URN: urn:nbn:se:uu:diva-306352DOI: 10.1007/s10959-012-0434-3ISI: 000334996100006OAI: oai:DiVA.org:uu-306352DiVA: diva2:1040450
Available from: 2016-10-27 Created: 2016-10-27 Last updated: 2016-10-27Bibliographically approved

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