GREM-like K processes on trees with infinite depth

TL;DR

This paper investigates the limit behavior of GREM-like K processes on infinitely deep trees, establishing the existence of a nontrivial infinite-level process and analyzing its properties.

Contribution

It introduces a method to derive the infinite-level limit of GREM-like processes on trees, under specific conditions ensuring martingality, and characterizes the limiting process's properties.

Findings

Established the existence of a nontrivial infinite-level limit process.

Derived an expression for the asymptotic empirical measure of cylinders.

Provided insights into the equilibrium measure of the infinite process.

Abstract

We take up the issue of deriving the limit as $n\to\infty$ of the GREM-like K process on a tree with $n$ levels. Under specific conditions on the parameters of the process, implying the martingality of a modification of the underlying clock process sequences, we obtain infinite level clock processes as nontrivial limits of the finite level clocks, and use them to construct a process on a suitable product space which is then shown to be the limit of the $n$ level K processes as $n\to\infty$. Some properties of the limiting, infinite level K process are established, like an expression for the asymptotic empirical measure of cylinders, giving information on the prospective equilibrium measure of the infinite level dynamics.