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

Abstract

We present a new perfect simulation algorithm for stationary chains having unbounded variable length memory. This is the class of infinite 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.

Stochastic Sequences with a Regenerative Structure that May Depend Both on the Future and on the Past

“…The following result is an immediate extension of Theorem 1 and the given conditions (15), (16), and (17).…”

“…First, there are directions where the methodology may be applied directly: Markov chains with long memory (see, e.g. [9], [11], and [16]); excited random walks (see, e.g. [4], [5], and [21]); modified random walks (see, e.g.…”

Perfect simulation for the infinite random cluster model, Ising and Potts models at low or high temperature

“…By (7) and the fact thatp e k+1 < u n,k+1 ≤ p e k+1 we have e k+1 ∈ H n . Let D be the connected component of H n containing e k+1 .…”

“…This paper has started some new research fields. One area of research concerns the Markov fields (see [3,12]); a second one concerns the processes with infinite memory (see [1,4,5,7]). Recently, these two areas of research have been in some sense unified by studying Gibbs measures with infinite interaction range (see [2,8]).…”

Infinite Systems of Interacting Chains with Memory of Variable Length—A Stochastic Model for Biological Neural Nets

“…A similar type of discontinuity has been considered in Gallo (2011) for stochastic chains with memory of variable length taking values in a finite alphabet.…”