Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:00:17.475Z Has data issue: false hasContentIssue false
  • English
  • Français

Concave Renewal Functions do not Imply DFR Interrenewal Times

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Yaming Yu
Yaming Yu*
Affiliation:
University of California
*
∗Postal address: Department of Statistics, University of California, Irvine, CA 92697, USA. Email address: yamingy@uci.edu
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.

Keywords

Renewal theorylog-convexity

MSC classification

Secondary: 60K05: Renewal theory
Type
Research Article
Information
Journal of Applied Probability , Volume 48 , Issue 2 , June 2011 , pp. 583 - 588
Copyright
Copyright © Applied Probability Trust 2011 

References

Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialized renewal processes. Ann. Prob. 8, 227–240.CrossRefGoogle Scholar
Brown, M. (1981). Further monotonicity properties for specialized renewal processes. Ann. Prob. 9, 891–895.Google Scholar
De Bruijn, N. G. and Erdős, P. (1953). On a recursion formula and some Tauberian theorems. J. Res. Nat. Bur. Standards 50, 161–164.Google Scholar
Hansen, B. G. (1988). On log-concave and log-convex infinitely divisible sequences and densities. Ann. Prob. 16, 1832–1839.Google Scholar
Hansen, B. G. and Frenk, J. B. G. (1991). Some monotonicity properties of the delayed renewal function. J. Appl. Prob. 28, 811–821.Google Scholar
Kaluza, T. (1928). Über die Koeffizienten reziproker Potenzreihen. Math. Z. 28, 161–170.CrossRefGoogle Scholar
Kebir, Y. (1997). Laplace transforms and the renewal equation. J. Appl. Prob. 34, 395–403.Google Scholar
Kijima, M. (1992). Further monotonicity properties of renewal processes. Adv. Appl. Prob. 24, 575–588.Google Scholar
Lund, R., Zhao, Y. and Kiessler, P. C. (2006). A monotonicity in reversible Markov chains. J. Appl. Prob. 43, 486–499.CrossRefGoogle Scholar
Shaked, M. and Zhu, H. (1992). Some results on block replacement policies and renewal theory. J. Appl. Prob. 29, 932–946.Google Scholar
Shanthikumar, J. G. (1988). DFR property of first-passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.Google Scholar
Szekli, R. (1986). On the concavity of the waiting-time distribution in some GI/G/1 queues. J. Appl. Prob. 23, 555–561.Google Scholar
Szekli, R. (1990). On the concavity of the infinitesimal renewal function. Statist. Prob. Lett. 10, 181–184.Google Scholar
Yu, Y. (2009). On the entropy of compound distributions on nonnegative integers. IEEE Trans. Inf. Theory 55, 3645–3650.CrossRefGoogle Scholar