Perfect simulation of autoregressive models with infinite memory

In this paper we consider the problem of determining the law of binary stochastic processes from transition kernels depending on the whole past. These kernels are linear in the past values of the process. They are allowed to assume values close to both 0 and 1, preventing the application of usual results on uniqueness. More precisely we give sufficient conditions for uniqueness and non-uniqueness. In the former case a perfect simulation algorithm is also given.Comment: 12 page

Attractive regular stochastic chains: perfect simulation and phase transition

We prove that uniqueness of the stationary chain, or equivalently, of the $g$-measure, compatible with an attractive regular probability kernel is equivalent to either one of the following two assertions for this chain: (1) it is a finitary coding of an i.i.d. process with countable alphabet, (2) the concentration of measure holds at exponential rate. We show in particular that if a stationary chain is uniquely defined by a kernel that is continuous and attractive, then this chain can be sampled using a coupling-from-the-past algorithm. For the original Bramson-Kalikow model we further prove that there exists a unique compatible chain if and only if the chain is a finitary coding of a finite alphabet i.i.d. process. Finally, we obtain some partial results on conditions for phase transition for general chains of infinite order.Comment: 22 pages, 1 pseudo-algorithm, 1 figure. Minor changes in the presentation. Lemma 6 has been remove

One-dimensional infinite memory imitation models with noise

In this paper we study stochastic process indexed by $\mathbb {Z}$ constructed from certain transition kernels depending on the whole past. These kernels prescribe that, at any time, the current state is selected by looking only at a previous random instant. We characterize uniqueness in terms of simple concepts concerning families of stochastic matrices, generalizing the results previously obtained in De Santis and Piccioni (J. Stat. Phys., 150(6):1017--1029, 2013).Comment: 22 pages, 3 figure

BACKWARD COALESCENCE TIMES FOR PERFECT SIMULATION OF CHAINS WITH INFINITE MEMORY

This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernandez and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, Fernandez and Ferrari (2002). We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events