Chains with unbounded variable length memory: perfect simulation and visible regeneration scheme

TL;DR

This paper introduces a perfect simulation algorithm for stationary chains with unbounded variable length memory, characterized by probabilistic context trees without continuity assumptions, extending renewal string chains with a visible regeneration scheme.

Contribution

It presents a novel perfect simulation method for a broad class of infinite memory chains defined by probabilistic context trees, without requiring continuity conditions.

Findings

The algorithm effectively simulates chains with unbounded variable length memory.

Chains exhibit a natural extension of renewal string chains with a visible regeneration scheme.

The approach broadens the scope of perfect simulation techniques for complex stochastic processes.

Abstract

We present a new perfect simulation algorithm for stationary chains having unbounded variable length memory. This is the class of infnite memory chains for which the family of transition probabilities is represented by a probabilistic context tree. We do not assume any continuity condition: our condition is expressed in terms of the structure of the context tree. More precisely, the length of the contexts is a deterministic function of the distance to the last occurrence of some determined string of symbols. It turns out that the resulting class of chains can be seen as a natural extension of the class of chains having a renewal string. In particular, our chains exhibit a visible regeneration scheme.