Stick-breaking, Clumping, and Occupation Laws
Connections are established among 'clumped' residual allocation models (RAMs), a generalized class of Dirichlet processes, and the occupation laws of certain discrete inhomogeneous Markov chains. An intermediate structure is introduced for a given RAM where probabilities associated to successive indices are added or clumped together to form another RAM. In particular, when the initial RAM has GEM distribution and clump boundaries are determined by the random times that an independent homogeneous Markov chain changes state, the joint law of the intermediate RAM and the locations visited is given in terms of a 'disordered' GEM sequence conditional on an induced homogeneous Markov chain. Through this joint law, an associated class of stick-breaking processes, including Dirichlet processes, are identified as the limits of empirical occupation measures of time-inhomogeneous Markov chains related to simulated annealing.