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Processes iterated ad libitum. (English) Zbl 1348.60026
Summary: Consider the \(n\)-th iterated Brownian motion \(I^{(n)} = B_n \circ \cdots \circ B_1\). N. Curien and T. Konstantopoulos [J. Theor. Probab. 27, No. 2, 433–448 (2014; Zbl 1331.60159)] proved that for any distinct numbers \(t_i \neq 0\), \((I^{(n)}(t_1), \ldots, I^{(n)}(t_k))\) converges in distribution to a limit \(I[k]\) independent of the \(t_i\)’s, exchangeable, and gave some elements on the limit occupation measure of \(I^{(n)}\). Here, we prove under some conditions that the finite-dimensional distributions of \(n\)-times iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of \(I[k]\) and of the finite-dimensional distributions of \(I^{(n)}\), as well as those of the iterated reflected Brownian motion iterated ad libitum.

60F05 Central limit and other weak theorems
60G52 Stable stochastic processes
60J65 Brownian motion
60G09 Exchangeability for stochastic processes
60G57 Random measures
60J05 Discrete-time Markov processes on general state spaces
60E99 Distribution theory
Full Text: DOI arXiv
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