Clumping, Stick-breaking, and an Inhomogeneous Markov Chain

Abstract

We study the connections among three types of objects: Residual Allocation Models (RAMs), a generalized class of stick-breaking processes which include Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains. These connections are established through a new clumping procedure on RAMs according to a clumping sequence. We introduce and analyze two methods of its application with respect to homogeneous Markov chains. Clumping according to the rst method results in a class of stick-breaking measures we later identify as the limit of the empirical occupation measures for particular time-inhomogeneous Markov chains. Clumping according to the second method leads to a general study of self-similarity as a characterization of random probability measures, which we apply in context. We further discuss the occupation law of an inhomogeneous Markov chain related to simulated annealing and its connections to stick-breaking measures. By introducing a reverse-chronological clumping procedure, we dene and study the local occupations up to time n of the Markov chain. As n gets large, these local occupations converge jointly to the clumped RAM structure resulting from the rst method above. We then explore additional properties of the generalized stick-breaking measure uncovered by the connection to the occupation law, and consider potential applications of results.