Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T11:37:25.985Z Has data issue: false hasContentIssue false
  • English
  • Français

Backward Coalescence Times for Perfect Simulation of Chains with Infinite Memory

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  04 February 2016

Emilio De Santis  and
Mauro Piccioni
Emilio De Santis*
Affiliation:
Sapienza Università di Roma
Mauro Piccioni*
Affiliation:
Sapienza Università di Roma
*
∗ Postal address: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
∗ Postal address: Dipartimento di Matematica, Sapienza Università di Roma, Piazzale Aldo Moro 2, 00185 Roma, Italy.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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 simulationcouplingchains with complete connections

MSC classification

Primary: 60G99: None of the above, but in this section 68U20: Simulation 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 2 , June 2012 , pp. 319 - 337
Copyright
© Applied Probability Trust 

References

Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
Comets, F., Fernández, R. and Ferrari, P. A. (2002). Processes with long memory: regenerative construction and perfect simulation. Ann. Appl. Prob. 12, 921–943.CrossRefGoogle Scholar
De Santis, E. and Piccioni, M. (2008). Exact simulation for discrete time spin systems and unilateral fields. Methodology Comput. Appl. Prob. 10, 105–120.CrossRefGoogle Scholar
Fernández, R. and Maillard, G. (2005). Chains with complete connections: general theory, uniqueness, loss of memory and mixing properties. J. Statist. Phys. 118, 555–588.CrossRefGoogle Scholar
Fernández, R., Ferrari, P. A. and Galves, A. (2001). Coupling, renewal and perfect simulations of chains of infinite order. Lecture Notes for the Vth Brazilian School of Probability, 86 pp.Google Scholar
Ferrari, P. A., Fernández, R. and Garcia, N. L. (2002). Perfect simulation for interacting point processes, loss networks and Ising models. Stoch. Process. Appl. 102, 63–88.CrossRefGoogle Scholar
Foss, S. and Konstantopoulos, T. (2003). Extended renovation theory and limit theorems for stochastic ordered graphs. Markov Process. Relat. Fields 9, 413–468.Google Scholar
Foss, S. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Commun. Statist. Stoch. Models 14, 187–203.CrossRefGoogle Scholar
Gallo, S. (2011). Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme. Adv. Appl. Prob. 43, 735–759.CrossRefGoogle Scholar
Gallo, S. and Garcia, N. L. (2010). Perfect simulation for stochastic chains of infinite memory: relaxing the continuity assumption. Preprint. Available at http://arXiv.org/abs/1005.5459v1.Google Scholar
Häggström, O. and Steif, J. E. (2000). Propp-Wilson algorithms and finitary codings for high noise Markov random fields. Combinatorics Prob. Comput. 9, 425–439.CrossRefGoogle Scholar
Møller, J. (2001). A review of perfect simulation in stochastic geometry. In Selected Proceedings of the Symposium on Inference for Stochastic Processes (Athens, GA, 2000; IMS Lecture Notes Monogr. Ser. 37), Institute of Mathematical Statistics, Beachwood, OH, pp. 333–355.CrossRefGoogle Scholar
Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25, 483–502.CrossRefGoogle Scholar
Propp, D. B. and Wilson, J. G. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9, 223–252.3.0.CO;2-O>CrossRefGoogle Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.CrossRefGoogle Scholar