  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.

Probability