Abstract:
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.

Abstract:
We consider the {following} oriented percolation model in $\mathbb{N}$. 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 $R_i$. Percolation occurs if every sites of $\mathbb{N}$ is covered by some interval. We obtain sharp conditions for both, positive and null probability of percolation. As corollaries, we extend results of the literature of rumor processes and discrete one-dimensional boolean percolation.

Abstract:
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.

Abstract:
In this paper, we study inference for chains of variable order under two distinct contamination regimes. Consider we have a chain of variable memory on a finite alphabet containing zero. At each instant of time an independent coin is flipped and if it turns head a contamination occurs. In the first regime a zero is read independent of the value of the chain. In the second regime, the value of another chain of variable memory is observed instead of the original one. Our results state that the difference between the transition probabilities of the original process and the corresponding ones of the contaminated process may be bounded above uniformly. Moreover, if the contamination probability is small enough, using a version of the Context algorithm we are able to recover the context tree of the original process through a contaminated sample.

Abstract:
Perfect simulation of an one-dimensional loss network on $\R$ with length distribution $\pi$ and cable capacity $C$ is performed using the clan of ancestors method. Domination of the clan of ancestors by a branching process with longer memory improves the sufficient conditions for the perfect scheme to be applicable.

Abstract:
Using the electric and coupling approaches, we derive a series of results concerning the mixing times for the stratified random walk on the d-cube, inspired in the results of Chung and Graham (1997) Stratified random walks on the n-cube.

Abstract:
Spatial birth and death processes are obtained as solutions of a system of stochastic equations. The processes are required to be locally finite, but may involve an infinite population over the full (noncompact) type space. Conditions are given for existence and uniqueness of such solutions, and for temporal and spatial ergodicity. For birth and death processes with constant death rate, a sub-criticality condition on the birth rate implies that the process is ergodic and converges exponentially fast to the stationary distribution.

Abstract:
We explicitly construct a coupling attaining Ornstein's $\bar{d}$-distance between ordered pairs of binary chains of infinite order. Our main tool is a representation of the transition probabilities of the coupled bivariate chain of infinite order as a countable mixture of Markov transition probabilities of increasing order. Under suitable conditions on the loss of memory of the chains, this representation implies that the coupled chain can be represented as a concatenation of iid sequence of bivariate finite random strings of symbols. The perfect simulation algorithm is based on the fact that we can identify the first regeneration point to the left of the origin almost surely.

Abstract:
we propose an adaptive method of analyzing a collection of curves which can be, individually, modeled as a linear combination of spline basis functions. through the introduction of latent bernoulli variables, the number of basis functions, the variance of the error measurements and the coefficients of the expansion are determined. we provide a modification of the stochastic em algorithm for which numerical results show that the estimates are very close to the true curve in the sense of l2 norm.

Abstract:
We propose an adaptive method of analyzing a collection of curves which can be, individually, modeled as a linear combination of spline basis functions. Through the introduction of latent Bernoulli variables, the number of basis functions, the variance of the error measurements and the coefficients of the expansion are determined. We provide a modification of the stochastic EM algorithm for which numerical results show that the estimates are very close to the true curve in the sense of L2 norm.