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

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.Comment: 27 pages, 10 figures, slight improvements of the results and simplification of the proof, simulations

On non-regular g-measures

We prove that g-functions whose set of discontinuity points has strictly negative topological pressure and which satisfy an assumption that is weaker than non-nullness, have at least one stationary g-measure. We also obtain uniqueness by adding conditions on the set of continuity points

Nonparametric statistical inference for the context tree of a stationary ergodic process

We consider the problem of estimating the context tree of a stationary ergodic process with finite alphabet without imposing additional conditions on the process. As a starting point we introduce a Hamming metric in the space of irreducible context trees and we use the properties of the weak topology in the space of ergodic stationary processes to prove that if the Hamming metric is unbounded, there exist no consistent estimators for the context tree. Even in the bounded case we show that there exist no two-sided confidence bounds. However we prove that one-sided inference is possible in this general setting and we construct a consistent estimator that is a lower bound for the context tree of the process with an explicit formula for the coverage probability. We develop an efficient algorithm to compute the lower bound and we apply the method to test a linguistic hypothesis about the context tree of codified written texts in European Portuguese

Discrete one-dimensional coverage process on a renewal process

We consider the {following} coverage model on $\mathbb{N}$. For each site $i\in \mathbb{N}$ we associate a pair $(\xi_i, R_i)$ where $\{\xi_0, \xi_1, \ldots \}$ is a 1-dimensional {undelayed} discrete renewal point process and $\{R_0,R_1,\ldots\}$ is an i.i.d. sequence of $\mathbb{N}$-valued random variables. At each site where $\xi_i=1$ we start an interval of length $R_i$. Coverage occurs if every site of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of coverage. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional Boolean percolation.Comment: 15 pages, 1 figur

Perfect simulation for locally continuous chains of infinite order

We establish sufficient conditions for perfect simulation of chains of infinite order on a countable alphabet. The new assumption, localized continuity, is formalized with the help of the notion of context trees, and includes the traditional continuous case, probabilistic context trees and discontinuous kernels. Since our assumptions are more refined than uniform continuity, our algorithms perfectly simulate continuous chains faster than the existing algorithms of the literature. We provide several illustrative examples.Comment: 38 pages, 1 figure, simplified proofs, improved results. We also removed the results concerning null chain

Frog models on trees through renewal theory

This paper studies a class of growing systems of random walks on regular trees, known as \emph{frog models with geometric lifetime} in the literature. With the help of results from renewal theory, we derive new bounds for their critical parameters. Our approach also improve the bounds of the literature for the critical parameter of a percolation model on trees called \emph{cone percolation}Comment: 11 pages, 1 figure, 2 table

Perfect simulation for stochastic chains of infinite memory: relaxing the continuity assumption

This paper is composed of two main results concerning chains of infinite order which are not necessarily continuous. The first one is a decomposition of the transition probability kernel as a countable mixture of unbounded probabilistic context trees. This decomposition is used to design a simulation algorithm which works as a combination of the algorithms given by Comets et al. (2002) and Gallo (2009). The second main result gives sufficient conditions on the kernel for this algorithm to stop after an almost surely finite number of steps. Direct consequences of this last result are existence and uniqueness of the stationary chain compatible with the kernel.Comment: 20 pages, 8 figures and 1 pseudo-code for the algorith

Markov Approximations of chains of infinite order in the dˉ\bar{d}-metric

We derive explicit upper bounds for the $\bar{d}$-distance between a chain of infinite order and its canonical $k$-steps Markov approximation. Our proof is entirely constructive and involves a "coupling from the past" argument. The new method covers non necessarily continuous probability kernels, and chains with null transition probabilities. These results imply in particular the Bernoulli property for these processes.Comment: 24 pages and 2 figures. Complete revision of the previous versio

Explicit estimates in the Bramson-Kalikow model

The aim of the present article is to explicitly compute parameters for which the Bramson-Kalikow model exhibits phase-transition. The main ingredient of the proof is a simple new criterion for non-uniqueness of $g$-measures. We show that the existence of multiple $g$-measures compatible with a function $g$ can be proved by estimating the $\bar{d}$-distances between some suitably chosen Markov chains. The method is optimal for the important class of binary regular attractive functions, which includes the Bramson-Kalikow model.Comment: The title in the previous version has an error. We also changed the structure of the article so that the main result now is the explicit criterion for phase transition of the BK process. The new title reflects this chang

Non-regular g-measures and variable length memory chains

It is well-known that there always exists at least one stationary measure compatible with a continuous g-function g. Here we prove that if the set of discontinuities of the g-function g has null measure under a candidate measure obtained by some asymptotic procedure, then this candidate measure is compatible with g. We explore several implications of this result, and discuss comparisons with the literature concerning assumptions and examples. Important part of the paper is concerned with the case of variable length memory chains, for which we obtain existence, uniqueness and weak-Bernoullicity (or $\beta$-mixing) under new assumptions. These results are specially designed for variable length memory models, and do not require vanishing uniform variation. We also provide a further discussion on some related notions, such as random context processes, non-essential discontinuities, and finally an example of everywhere discontinuous stationary measure.Comment: 33 page