Chains with Complete Connections: General Theory, Uniqueness, Loss of Memory and Mixing Properties

Abstract

We introduce an statistical mechanical formalism for the study of discrete-time stochastic processes with which we prove: (i) General properties of extremal chains, including triviality on the tail σ-algebra, short-range correlations, realization via infinite-volume limits and ergodicity. (ii) Two new sufficient conditions for the uniqueness of the consistent chain. The first one is a transcription of a criterion due to Georgii for one-dimensional Gibbs measures, and the second one corresponds to Dobrushin criterion in statistical mechanics. (iii) Results on loss of memory and mixing properties for chains in the Dobrushin regime. These results are complementary of those existing in the literature, and generalize the Markovian results based on the Dobrushin ergodic coefficient.

Explicit estimates in the Bramson–Kalikow model

“…Known criteria for uniqueness (Johansson et al, 2012;Fernández & Maillard, 2005) don't give such conditions. Therefore, the existence of a sharp transition from uniqueness to nonuniqueness regime for the BK model still remains an interesting open problem.…”

“…They constitute an important class of stochastic models, which includes, for example, Markov chains, stochastic models that exhibit nonuniqueness and models that are not Gibbsian (Fernández et al, 2011). The question of uniqueness of g-measures was extensively studied and important progresses have been obtained in several areas related to probability and ergodic theory, from the seminal works of Onicescu & Mihoc (1935); Doeblin & Fortet (1937) to recent advances in Johansson et al (2012); Gallo & Paccaut (2013), and the contributions of Harris (1955); Keane (1972); Walters (1975); Lalley (1986); Stenflo (2003); Fernández & Maillard (2005) among many others. Notwithstanding, the problem of non-uniqueness is much less understood and the literature is still based on few examples (Bramson & Kalikow, 1993;Hulse, 2006;Berger et al, 2005).…”

Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory

“…This example does not satisfy the sufficient condition for uniqueness of a compatible process given in Proposition 4.2 of [4].…”

Correction to ‘A class of unique g -measures’

“…The referee has pointed out a recent paper by Fernández and Maillard [3] in which they study non-stationary chains with complete connections. They prove the uniqueness of such chains subject to a hypothesis which, in the present context, is equivalent to that of Theorem 1.1.…”