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

On the uniqueness of kernels

AbstractLet S be a compact orientable surface. For any graph G embedded on S and any closed curve D on S we define μG(D) as the minimum number of intersections of G and D′, where D′ ranges over all closed curves freely homotopic to D. We call G a kernel if μG′ ≠ μG for each proper minor G′ of G. We prove that if G and G′ are kernels with μG = μG′ (in such a way that each face of G is an open disk), then G′ can be obtained from G by a series of the following operations: (i) homotopic shifts over S; (ii) taking the surface dual graph; (iii) ΔY-exchange (i.e., replacing a vertex v of degree 3 by a triangle connecting the three vertices adjacent to v, or conversely)

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