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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.

##### MSC:
 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
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