Processes iterated ad libitum.

Consider the n th iterated Brownian motion I ( n ) = B n ∘ ⋯ ∘ B 1 . Curien and Konstantopoulos proved that for any distinct numbers t i ≠ 0 , ( I ( n ) ( t 1 ) , … , 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, finite dimensional distributions of n th iterated two-sided stable processes converge, and the same holds the reflected Brownian motions. We give a description of the law of I [ k ] , of the finite dimensional distributions of I ( n ) , as well as those of the iterated reflected Brownian motion iterated ad libitum. [ABSTRACT FROM AUTHOR]