Processes iterated ad libitum

TL;DR

This paper studies the limiting behavior of iterated stochastic processes, including Brownian and stable processes, and characterizes their finite-dimensional distributions and occupation measures.

Contribution

It extends previous results by proving convergence of finite-dimensional distributions for iterated stable and reflected Brownian processes and describes their laws.

Findings

Finite-dimensional distributions of iterated stable processes converge.

Finite-dimensional distributions of iterated reflected Brownian motions converge.

Provides explicit descriptions of the laws of the limit processes.

Abstract

Consider the $n$th iterated Brownian motion $I^{(n)}=B_n \circ\cdots \circ B_1$. Curien and Konstantopoulos proved that for any distinct numbers $t_i\neq 0$, $(I^{(n)}(t_1),\dots,I^{(n)}(t_k))$ 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.