Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-28T19:41:10.525Z Has data issue: false hasContentIssue false
  • English
  • Français

An infinite particle system with continual input and random death

Published online by Cambridge University Press:  01 July 2016

J. N. McDonald  and
N. A. Weiss
J. N. McDonald
Affiliation:
Arizona State University
N. A. Weiss
Affiliation:
Arizona State University
Get access Rights & Permissions [Opens in a new window]

Abstract

At times n = 0, 1, 2, · · · a Poisson number of particles enter each state of a countable state space. The particles then move independently according to the transition law of a Markov chain, until their death which occurs at a random time. Several limit theorems are then proved for various functionals of this infinite particle system. In particular, laws of large numbers and central limit theorems are proved.

Keywords

INFINITE PARTICLE SYSTEMSMARKOV CHAINSLAWS OF LARGE NUMBERSCENTRAL LIMIT THEOREMSRANDOM LIFETIMES
Type
Research Article
Information
Advances in Applied Probability , Volume 10 , Issue 4 , December 1978 , pp. 764 - 787
Copyright
Copyright © Applied Probability Trust 1978 

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] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities, 2nd edn. Springer-Verlag, New York.Google Scholar
[2] Chung, K. L. (1974) A Course in Probability Theory, 2nd edn. Academic Press, New York.Google Scholar
[3] Derman, C. (1955) Some contributions to the theory of denumerable Markov chains. Trans. Amer. Math. Soc. 79, 541–555.CrossRefGoogle Scholar
[4] Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd edn. Wiley, New York.Google Scholar
[5] Jacobs, P. A. (1974) Functionals of an infinite particle system with independent motions, creations and annihilation. Adv. Appl. Prob. 6, 636–650.Google Scholar
[6] Knopp, K. (1951) Theory and Application of Infinite Series. Blackie, London.Google Scholar
[7] Leysieffer, F. W. (1970) A system of Markov chains with random life times. Ann. Math. Statist. 41, 576–584.CrossRefGoogle Scholar
[8] Port, S. C. (1965) Limit theorems involving capacities for recurrent Markov chains. J. Math. Anal. Appl. 12, 555–569.Google Scholar
[9] Port, S. C. (1966) A system of denumerably many transient Markov chains. Ann. Math. Statist. 37, 406–411.Google Scholar
[10] Port, S. C. (1966) Limit theorems involving capacities. J. Math. Mech. 15, 805–832.Google Scholar
[11] Spitzer, F. (1964) Principles of Random Walk. Van Nostrand, Princeton, N.J.Google Scholar
[12] Weiss, N. A. (1971) Limit theorems for infinite particle systems. Z. Wahrscheinlichkeitsth. 20, 87–101.Google Scholar
[13] Weiss, N. A. (1972) The occupation time of a set by countably many recurrent random walks. Ann. Math. Statist. 43, 293–302.Google Scholar