# Stick-breaking processes, clumping, and Markov chain occupation laws

## TL;DR

This paper explores the relationships between clumped residual allocation models, a broad class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain Markov chains, revealing new connections and limit behaviors.

## Contribution

It introduces an intermediate structure in RAMs involving clumping, linking stick-breaking processes to Markov chain occupation laws, and characterizes their limits in new settings.

## Key findings

- Joint law of intermediate RAM and visited states expressed via disordered GEM sequence.
- Identifies a class of stick-breaking processes as limits of empirical occupation measures.
- Connects inhomogeneous Markov chain behavior with generalized stick-breaking processes.

## Abstract

We consider the connections among `clumped' residual allocation models (RAMs), a general class of stick-breaking processes including Dirichlet processes, and the occupation laws of certain discrete space time-inhomogeneous Markov chains related to simulated annealing and other applications. An intermediate structure is introduced in a given RAM, where proportions between successive indices in a list are added or clumped together to form another RAM. In particular, when the initial RAM is a Griffiths-Engen-McCloskey (GEM) sequence and the indices are given by the random times that an auxiliary Markov chain jumps away from its current state, the joint law of the intermediate RAM and the locations visited in the sojourns is given in terms of a `disordered' GEM sequence, and an induced Markov chain. Through this joint law, we identify a large class of `stick breaking' processes as the limits of empirical occupation measures for associated time-inhomogeneous Markov chains.