GREM-like K processes on trees with infinite depth
  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. 
Luiz Renato Fontes and Gabriel R. C. Peixoto
math.PR