Processes iterated ad libitum

Consider the nth iterated Brownian motion I(n)=Bn∘⋯∘B1. Curien and Konstantopoulos proved that for any distinct numbers ti≠0, (I(n)(t1),…,I(n)(tk)) converges in distribution to a limit I[k] independent of the ti's, exchangeable, and gave some elements on the limit occupation measure of I(n). Here, we prove under some conditions, finite dimensional distributions of nth 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.