Journal of Applied Probability

Backward coalescence times for perfect simulation of chains with infinite memory

Emilio De Santis and Mauro Piccioni

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Appl. Probab., Volume 49, Number 2 (2012), 319-337.

Dates
First available in Project Euclid: 16 June 2012

Permanent link to this document
https://projecteuclid.org/euclid.jap/1339878789

Digital Object Identifier
doi:10.1239/jap/1339878789

Mathematical Reviews number (MathSciNet)
MR2977798

Zentralblatt MATH identifier
1246.60079

Subjects
Primary: 60G99: None of the above, but in this section 68U20: Simulation [See also 65Cxx] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)

Keywords
Perfect simulation coupling chains with complete connections

Citation

De Santis, Emilio; Piccioni, Mauro. Backward coalescence times for perfect simulation of chains with infinite memory. J. Appl. Probab. 49 (2012), no. 2, 319--337. doi:10.1239/jap/1339878789. https://projecteuclid.org/euclid.jap/1339878789


Export citation

References

  • Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.
  • Comets, F., Fernández, R. and Ferrari, P. A. (2002). Processes with long memory: regenerative construction and perfect simulation. Ann. Appl. Prob. 12, 921–943.
  • De Santis, E. and Piccioni, M. (2008). Exact simulation for discrete time spin systems and unilateral fields. Methodology Comput. Appl. Prob. 10, 105–120.
  • 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.
  • 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, 86pp.
  • 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.
  • Foss, S. and Konstantopoulos, T. (2003). Extended renovation theory and limit theorems for stochastic ordered graphs. Markov Process. Relat. Fields 9, 413–468.
  • Foss, S. and Tweedie, R. L. (1998). Perfect simulation and backward coupling. Commun. Statist. Stoch. Models 14, 187–203.
  • Gallo, S. (2011). Chains with unbounded variable length memory: perfect simulation and a visible regeneration scheme. Adv. Appl. Prob. 43, 735–759.
  • 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.
  • 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.
  • 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.
  • Murdoch, D. J. and Green, P. J. (1998). Exact sampling from a continuous state space. Scand. J. Statist. 25, 483–502.
  • 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.
  • Williams, D. (1991). Probability with Martingales. Cambridge University Press.