Abstract
We show convergence in variation to a unique stationary state for a class of point processes (respectively, stochastic sequences) with stochastic intensity kernels (respectively, transition probabilities) including the (A,m)-processes of Lindvall [12]. This is done under two basic conditions: first, the random memory of the processes considered is consistent or non-reusable (that is, past information not used at a given time cannot be recalled at a later time) and secondly, the kernels have a deterministic fixed component for which the memory is almost surely finite.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen,Applied Probability and Queues (Wiley, 1987).
S. Asmussen, On coupling and weak convergence to stationarity, Ann. Appl. Prob. 2 (1992) 739–751.
F. Baccelli and P. Brémaud,Elements of Queueing Theory (Springer, New York, 1994).
A. Borovkov,Asymptotic Methods in Queueing Theory (Wiley, New York, 1984).
A. Brandt, P. Franken and B. Lisek,Stationary Stochastic Models (Wiley, New York, 1992).
D. Daley and D. Vere-Jones,An Introduction to the Theory of Point Processes (Springer, New York, 1988).
M. Delasnerie, Flot mélangeant et mesures de Palm, Ann. Inst. Henri Poincaré 13 (1977) 357–369.
A.A. Borovkov and S.G. Foss, Stochastically recursive sequences and their generalizations, Sib. Adv. Math. 2 (1992) 16–81.
J. Jacod, Multivariate point processes: predictable projections, Radon-Nikodym derivatives, representation of martingales, Z. Wahrs. 31 (1975) 235–253.
S.P. Lalley, Renewal theorem for a class of stationary sequences, Prob. Theory Rel. Fields 72 (1986) 195–213.
T. Lindvall, On coupling of renewal processes with use of failure rates, Stoch. Proc. Appl. 22 (1986) 1–15.
T. Lindvall, Ergodicity and inequalities in a class of point processes, Stoch. Proc. Appl. 30 (1988) 121–131.
T. Lindvall,Lectures on Coupling Methods (Wiley, New York, 1992).
A. Skorohod, On a representation of random variables, Theory Prob. Appl. 21 (1976) 628–632.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Brémaud, P., Massoulié, L. Imbedded construction of stationary sequences and point processes with a random memory. Queueing Syst 17, 213–234 (1994). https://doi.org/10.1007/BF01158695
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01158695