Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory

Abstract

This paper is devoted to the perfect simulation of a stationary process with an at most countable state space. The process is specified through a kernel, prescribing the probability of the next state conditional to the whole past history. We follow the seminal work of Comets, Fernández and Ferrari (2002), who gave sufficient conditions for the construction of a perfect simulation algorithm. We define backward coalescence times for these kind of processes, which allow us to construct perfect simulation algorithms under weaker conditions than in Comets, . We discuss how to construct backward coalescence times (i) by means of information depths, taking into account some a priori knowledge about the histories that occur; and (ii) by identifying suitable coalescing events.

Perfect Simulation of Autoregressive Models with Infinite Memory

“…For this general algorithm to work, in the case ∞ k=1 |θ k | < 1, a control on the tails of the distribution θ * is needed, which is quite stronger than that required by Theorem 4. We should also mention that in [DSP12] a modification of such an algorithm is proposed, which works also for an example in which θ 0 = 0 and ∞ k=1 |θ k | = 1. But such a modification was rather ad hoc, and it was not clear how to pursue the same strategy for a generic vector θ with ∞ k=1 |θ k | = 1.…”

Attractive regular stochastic chains: perfect simulation and phase transition

“…As an example of application of the above result, if a probability kernel has summable continuity rate var k , we can construct a CFTP algorithm with stopping time θ such that E[θ] ≤ C ∞ k=1 var k , where C is a numerical constant [5]. We refer the reader to [11], [12], and [6] to obtain bounds on probability of θ for different conditions and not necessary regular kernels. Now we have the last ingredient for the proof of Theorem 1.…”

Perfect simulation for locally continuous chains of infinite order

“…Let us say, before going into any further details, that the update function we will use is the same (with some simple changes to make it suitable when P is not necessarily continuous) as the one used by Comets et al (2002), and which underlies the works of Gallo (2011) andPiccioni (2012). This update function is defined through the partition of [0, 1[ represented on Figure 1, where for any a and a, the intervals have length…”

“…Other recent results in the area are the papers of Garivier (2011) andPiccioni (2012). The former introduced an elegant CFTP algorithm which works without the weak non-nullness assumption, designedà la Propp & Wilson (1996).…”