J. Appl. Prob. 49, 319–337 (2012) Printed in England  Applied Probability Trust 2012 BACKWARD COALESCENCE TIMES FOR PERFECT SIMULATION OF CHAINS WITH INFINITE MEMORY EMILIO DE SANTIS ∗ ∗∗ and MAURO PICCIONI, ∗ ∗∗∗ Sapienza Università di Roma 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, Fernández and Ferrari (2002). 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. Keywords: Perfect simulation; coupling; chains with complete connections 2010 Mathematics Subject Classiﬁcation: Primary 60G99; 68U20; 60J10 1. Introduction and main deﬁnitions Perfect simulation algorithms for stochastic processes have been developed mostly for Markov chains, starting from the original coupling-from-the-past algorithm presented in the founding paper [14]. Later, the fundamental role of the so-called stochastic recursive sequences for perfect simulation was recognized [8]. In [13] a stochastic recursive sequence for perfect simulation was constructed, called the gamma coupler, for transition kernels satisfying a minorization condition. We also mention the area of perfect simulation devoted to spatial contexts, such as stochastic geometry [6], [12] and random ﬁelds [3], [11]. In [2] the ideas of coupling from the past have been extended to processes constructed through a transition kernel depending on the whole past, leading to the construction of a perfect simulation algorithm. A somewhat more general presentation was given in [5]. We need a greater generality to cover a number of different transition kernels. We also mention the paper [7], which does not refer to a kernel, but only to a process and its ‘shifted’ versions, the shift being associated to a ‘stationary environment’ modeled in the probability space. Throughout the paper, we consider stochastic processes deﬁned on Z, with values in a ﬁnite or countable alphabet G. The law of the process is speciﬁed through a transition kernel, prescribing the probability that each individual letter of the alphabet occurs, conditional to the whole history preceding it. Received 22 February 2011; revision received 13 September 2011. ∗ Postal address: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy. ∗∗ Email address: desantis@mat.uniroma1.it ∗∗∗ Email address: piccioni@mat.uniroma1.it 319 https://doi.org/10.1239/jap/1339878789 Published online by Cambridge University Press