Perfect simulation for locally continuous chains of infinite order

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.

Stochastic Processes With Random Contexts: A Characterization and Adaptive Estimators for the Transition Probabilities

“…By Proposition 5, L(x) < +∞ and p(a|x) = p − (a|x −1 −L(x) ) P past -almost-surely. Recalling that δ = n −2 , (19) in Subsection VII-A4 may be used to show that P (E) ≥ 1 − n −2 , where:…”

One-Dimensional Infinite Memory Imitation Models with Noise

“…By taking Cesaro averages on a window increasing to Z and going to the limit one gets at least one stationary process with single site marginals still equal to λ. For 3., let X ∈ G(p) be stationary and call λ its single-site marginals: then, similarly to (14) λ(g) = P (X n = g) = E(P (X n = g | X i , i < n)) = k∈A θ k (λP (k) )(g) = (λP )(g).…”

“…The mixture decomposition presented in [4] is not unique. Other decompositions have been proposed to prove uniqueness [8,14,13], leading to relax not only the regularity but also the positivity assumption in [4].…”

Non-regular g-measures and variable length memory chains

“…Question 1 and 2 above were answered, in both literatures, essentially under the assumption that the g-function is continuous (this implies existence), strictly positive, and eventually with a rapidly vanishing variation if we wish to have uniqueness. In the last 10 years, several works (De Santis & Piccioni, 2012;Gallo & Garcia, 2013;Gallo & Paccaut, 2013;Oliveira, 2015) have studied the case of possibly discontinuous g-functions (kernels) under several perspectives, including the existence problem, the uniqueness problem and further properties such as perfect simulation, mixing properties, statistical inference. The present paper is more in the vein (and indeed can be considered a sequel) of Gallo & Paccaut (2013).…”

“…On the theoretical side, Gallo (2011) seems to be the first work interested in this model under the perspective of stochastic processes. This class has been also used as a tool to study general g-measures, for instance in Gallo & Garcia (2013); Gallo & Paccaut (2013); Garivier (2015); Oliveira (2015). Finally, some recent works have explored the relation with dynamical systems (Cénac et al, 2012; or used this class to create interesting random walk models (Cénac et al, , 2019.…”