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Processes iterated ad libitum

Research paper by Jérôme Casse, Jean-François Marckert

Indexed on: 11 Jun '16Published on: 06 Jun '16Published in: Stochastic Processes and their Applications



Abstract

Consider the nnth iterated Brownian motion I(n)=Bn∘⋯∘B1I(n)=Bn∘⋯∘B1. Curien and Konstantopoulos proved that for any distinct numbers ti≠0ti≠0, (I(n)(t1),…,I(n)(tk))(I(n)(t1),…,I(n)(tk)) converges in distribution to a limit I[k]I[k] independent of the titi’s, exchangeable, and gave some elements on the limit occupation measure of I(n)I(n). Here, we prove under some conditions, finite dimensional distributions of nnth iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of I[k]I[k], of the finite dimensional distributions of I(n)I(n), as well as those of the iterated reflected Brownian motion iterated ad libitum.