Perfect simulation for stochastic chains of infinite memory: relaxing the continuity assumption

TL;DR

This paper introduces a novel decomposition and simulation algorithm for infinite memory chains without the need for continuity, establishing conditions for almost sure termination and ensuring existence and uniqueness of the stationary chain.

Contribution

It provides a new decomposition of transition kernels and a simulation method applicable to non-continuous chains, relaxing previous continuity assumptions.

Findings

Decomposition of transition kernels into countable mixtures of probabilistic context trees

A simulation algorithm that combines existing methods for infinite memory chains

Conditions under which the simulation algorithm terminates almost surely

Abstract

This paper is composed of two main results concerning chains of infinite order which are not necessarily continuous. The first one is a decomposition of the transition probability kernel as a countable mixture of unbounded probabilistic context trees. This decomposition is used to design a simulation algorithm which works as a combination of the algorithms given by Comets et al. (2002) and Gallo (2009). The second main result gives sufficient conditions on the kernel for this algorithm to stop after an almost surely finite number of steps. Direct consequences of this last result are existence and uniqueness of the stationary chain compatible with the kernel.