Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T17:41:23.617Z Has data issue: false hasContentIssue false
  • English
  • Français

General solidarity theorems for semi-Markov processes

Published online by Cambridge University Press:  14 July 2016

Choong K. Cheong  and
Jozef L. Teugels
Choong K. Cheong*
Affiliation:
Catholic University of Louvain, Belgium
Jozef L. Teugels
Affiliation:
Catholic University of Louvain, Belgium
*
*Now at the University of Malaya, Kuala Lumpur.
Get access Rights & Permissions [Opens in a new window]

Abstract

Let {Zt, t ≧ 0} be an irreducible regular semi-Markov process with transition probabilities Pij (t). Let f(t) be non-negative and non-decreasing to infinity, and let λ ≧ 0. This paper identifies a large set of functions f(t) with the solidarity property that convergence of the integral ≧ eλtf(t)Pij(t) dt for a specific pair of states i and j implies convergence of the integral for all pairs of states. Similar results are derived for the Markov renewal functions Mij (t). Among others it is shown that f(t) can be taken regularly varying.

Keywords

SEMI-MARKOV PROCESSMARKOV RENEWAL PROCESSSOLIDARITY PROPERTYCLASS PROPERTYREGULAR VARIATIONSLOWLY VARYING FUNCTIONLIMIT THEOREMSGEOMETRIC ERGODICITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 9 , Issue 4 , December 1972 , pp. 789 - 802
Copyright
Copyright © Applied Probability Trust 1972 

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] Callaert, H. and Teugels, J. L. (1972) Regular variation of Markov renewal functions. (To appear.) Google Scholar
[2] Cheong, C. K. (1968) Ergodic and ratio limit theorems for a-recurrent semi-Markov processes. Zeit. Wahrscheinlichkeitsth. 9, 270–286.Google Scholar
[3] Chung, K. L. (1967) Markov Chains with Stationary Transition Probabilities. Springer-Verlag, New York.Google Scholar
[4] Feller, W. (1966) An Introduction to the Theory of Probability and its Applications. Vol. II. J. Wiley, New York.Google Scholar
[5] Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties and Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1231–1259.CrossRefGoogle Scholar
[6] Teugels, J. L. (1968) Exponential ergodicity in Markov renewal processes. J. Appl. Prob. 5, 387–400.Google Scholar
[7] Teugels, J. L. (1970) Regular variation of Markov renewal functions. J. London Math. Soc. II, 2, 179–190.CrossRefGoogle Scholar