Processes with Long Memory: Regenerative Construction and Perfect Simulation

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 …

Perfect Simulation Of Processes With Long Memory: A `Coupling Into And From The Past' Algorithm

We describe a new algorithm for the perfect simulation of variable length Markov chains and random systems with perfect connections. This algorithm, which generalizes Propp and Wilson's simulation scheme, is based on the idea of coupling into and from the past. It improves on existing algorithms by relaxing the conditions on the kernel and by accelerating convergence, even in the simple case of finite order Markov chains. Although chains of variable or infinite order have been widely investigated for decades, their use in applied probability, from information theory to bio-informatics and ling…

Perfect simulation for interacting Hawkes processes with variable length memory

We consider a nonlinear multivariate Hawkes process having a variable length memory which allows to describe the activity of a neuronal network by its membrane potential. We propose a graphical construction of the process and we construct, by means of a perfect simulation algorithm, a stationary version of the process. By making the hypothesis that the spiking rate $β_i$ of the neuron $i \in I $ is bounded, we construct an algorithm based on a priori realizations of the Poisson process $(M^i, i \in I)$. We show that there exists a critical value $δ_c$ such that if $\underlineδ > δ_c$ (where $\…

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 …

对无限记忆链完美模拟的新审视

A new look at perfect simulation for chains with infinite memory

针对无限记忆链的完美模拟问题，提出两种支持未知过去符号、可处理稀疏依赖的过去耦合型完美模拟算法，给出有限停止及平均步数有限的条件，并验证了其适用范围优于现有结果。

Improved bounds for perfect simulation of a continuous one-dimensional loss network

Perfect simulation of an one-dimensional loss network on $\R$ with length distribution $π$ 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.

相互作用Hawkes过程的完美模拟：重置诱导的变量长度记忆

Perfect simulation for interacting Hawkes processes with reset-induced variable length memory

本文研究了具有重置机制的相互作用非线性Hawkes过程，通过图构造和祖先族方法，证明了子临界条件下的有限祖先族，从而构建了平稳解并提出精确的完美模拟算法。

Simple perfect samplers using monotone birth-and-death processes

This paper proposes simple perfect samplers using monotone birth-and-death processes (BD-processes), which draw samples from an arbitrary finite discrete target distribution. We first construct a monotone BD-process whose stationary distribution is equal to the target distribution. We then derive upper bounds for the expected coalescence time of the copies of the monotone BD-process. We also establish upper bounds for the expected values and tail probabilities of the running times of two perfect samplers, which are Doubling CFTP and Read-once CFTP using our monotone BD-process. The latter samp…

Perfect simulation of a coupling achieving the $\bar{d}$-distance between ordered pairs of binary chains of infinite order

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…

Hawkes processes with variable length memory and an infinite number of components

In this paper, we build a model for biological neural nets where the activity of the network is described by Hawkes processes having a variable length memory. The particularity of this paper is to deal with an infinite number of components. We propose a graphical construction of the process and we build, by means of a perfect simulation algorithm, a stationary version of the process. To carry out this algorithm, we make use of a Kalikow-type decomposition technique. Two models are described in this paper. In the first model, we associate to each edge of the interaction graph a saturation thres…