On asymptotic behavior of iterates of piecewise constant monotone maps
arXiv:2110.09731
Abstract
In this paper we study the rate of convergence of the iterates of \iid random piecewise constant monotone maps to the time- transport map for the process of coalescing Brownian motions. We prove that the rate of convergence is given by a power law. The time-1 map for the coalescing Brownian motions can be viewed as a fixed point for a natural renormalization transformation acting in the space of probability laws for random piecewise constant monotone maps. Our result implies that this fixed point is exponentially stable.