A new look at perfect simulation for chains with infinite memory

TL;DR

This work develops two novel perfect simulation algorithms for chains with infinite memory by expanding the coupling from the past framework to include unknown past states. The first algorithm permits spontaneous symbol occurrences (β>0), while the second handles β=0 cases via a finite-memory, multi-trajectory coupling, with rigorous conditions on long-past decay ensuring almost-sure finite stopping times and, under stronger decay, finite mean running times. The authors show these algorithms generate samples from the unique stationary measure, entail loss of memory, regeneration, and concentration of measure, and apply to a broad class of kernels—including sparse past dependencies—where previous results may fail. Through extensive examples and comparative discussion, the paper positions these methods as a flexible, generalizable approach to perfect simulation for infinite-memory processes, with clear theoretical guarantees and practical regeneration implications.

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.

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Theorems & Definitions (39)

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