A new look at perfect simulation for chains with infinite memory

TL;DR

This paper introduces two novel perfect simulation algorithms for chains with infinite memory, accommodating unknown states and sparse dependencies, with guarantees on finite stopping time and efficiency.

Contribution

The paper presents new coupling of past algorithms for infinite memory chains that handle unknown states and sparse dependencies, with proven finite stopping times.

Findings

Algorithms stop after finite steps almost surely.

Mean number of steps is finite under certain conditions.

Applicable to chains with long-range dependencies and unknown states.

Abstract

In this article we introduce two new perfect simulation algorithms for chains with infinite memory. Both algorithms belong to the coupling of past procedures. The novelty of our approach is that it allows to include unknown states to the possible past symbols such that we can also deal with sparsely distributed past dependencies. In our first algorithm, spontaneous occurrence of symbols is possible. This means that there is a positive probability that the chain chooses the next symbol independently of the past. Our second algorithm deals with the case in which spontaneous occurrence of symbols is not possible. Chains with infinite memory are discrete-time stochastic processes in which the distribution of the next symbol depends on all past symbols. These transition probabilities are described by a probability kernel. Our results give conditions on the way the dependency of the transition kernel on long past strings decays, guaranteeing that our algorithms stop after a finite number of steps almost surely. Strengthening these conditions, we show that the mean number of steps of our algorithms is finite. We discuss the consequence of having a coupling from the past algorithm with such properties and we present examples in which our results can be applied while others result in the literature cannot be applied.