Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-27T05:00:07.029Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of interacting marked point processes: rate of convergence to equilibrium

Part of: Stochastic processes

Published online by Cambridge University Press:  14 July 2016

G. L. Torrisi
G. L. Torrisi*
Affiliation:
Istituto per le Applicazioni del Calcolo, Roma
*
∗ Postal address: Istituto per le Applicazioni del Calcolo ‘M. Picone’ (IAC-CNR), Viale del Policlinico 137, 00161 Roma, Italy. Email address: torrisi@iac.rm.cnr.it
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we obtain the rate of convergence to equilibrium of a class of interacting marked point processes, introduced by Kerstan, in two different situations. Indeed, we prove the exponential and subexponential ergodicity of such a class of stochastic processes. Our results are an extension of the corresponding results of Brémaud, Nappo and Torrisi. The generality of the dynamics which we take into account allows the application to the so-called loss networks, and multivariate birth and death processes.

Keywords

Point processesstochastic intensityergodicityHawkes processesloss networksbirth and death processes

MSC classification

Primary: 60G55: Point processes
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 137 - 160
Copyright
Copyright © Applied Probability Trust 2002 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]. Asmussen, S. (1987). Applied Probability and Queues. John Wiley, Chichester.Google Scholar
[2]. Asmussen, S., Henriksen, F. L. and Klüppelberg, C. (1994). Large claims approximations for risk processes in a Markovian environment. Stoch. Proc. Appl. 54, 29–43.Google Scholar
[3]. Athreya, K. B., and Ney, P. E. (1972). Branching Processes. Springer, New York.CrossRefGoogle Scholar
[4]. Brémaud, P. (1981). Point Processes and Queues. Springer, New York.CrossRefGoogle Scholar
[5]. Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 1563–1588.CrossRefGoogle Scholar
[6]. Brémaud, P., Nappo, G., and Torrisi, G. L. (2002). Rate of convergence to equilibrium of marked Hawkes processes. J. Appl. Prob. 39, 123–136.Google Scholar
[7]. Daley, D. J., and Vere-Jones, D. (1988). An Introduction to the Theory of Point Processes. Springer, New York.Google Scholar
[8]. Hawkes, A. G., and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493–503.Google Scholar
[9]. Jacod, J. (1975). Multivariate point processes: predictable projections, Radon–Nicodym derivatives, representations of martingales. Z. Wahrscheinlichkeitsth. 31, 235–253.Google Scholar
[10]. Kerstan, J. (1964). Teilprozesse Poissonscher Prozesse. In Trans. 3rd Prague Conf. Information Theory, Statistical Decision Functions, Random Processes (Liblice, 1962), Czechoslovak Academy of Science, Prague, pp. 377–403.Google Scholar
[11]. Lancaster, P., and Tismenetski, M. (1985). The Theory of Matrices. Academic Press, New York.Google Scholar
[12]. Last, G. (1996). Coupling with compensators. Stoch. Proc. Appl. 65, 147–170.CrossRefGoogle Scholar
[13]. Lindvall, T. (1992). Lectures on the Coupling Method. John Wiley, New York.Google Scholar
[14]. Massoulié, L. (1995). Stabilité, Simulation et Optimisation de Systèmes à Evénements Discrets. Doctoral Thesis, Université de Paris-sud.Google Scholar
[15]. Massoulié, L. (1998). Stability results for a general class of interacting point processes dynamics, and applications. Stoch. Proc. Appl. 75, 1–30.Google Scholar
[16]. Ogata, Y. (1981). On Lewis’ simulation method for point processes. IEEE Trans. Inf. Theory 27, 23–31.Google Scholar