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April, 1980 Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes
Mark Brown
Ann. Probab. 8(2): 227-240 (April, 1980). DOI: 10.1214/aop/1176994773


Renewal processes with increasing mean residual life and decreasing failure rate interarrival time distributions are investigated. Various two-sided bounds are obtained for $M(t)$, the expected number of renewals in $\lbrack 0, t\rbrack$. It is shown that if the interarrival time distribution has increasing mean residual life with mean $\mu$, then the expected forward recurrence time is increasing in $t \geqslant 0$, as is $M(t) - t/\mu$. If the interarrival time distribution has decreasing failure rate then $M(t)$ is concave, and the forward and backward recurrence time distributions are stochastically increasing in $t \geqslant 0$.


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Mark Brown. "Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes." Ann. Probab. 8 (2) 227 - 240, April, 1980.


Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0429.60084
MathSciNet: MR566590
Digital Object Identifier: 10.1214/aop/1176994773

Primary: 60K05
Secondary: 60699

Keywords: almost sure constructions , bounds and inequalities for stochastic processes , forward and backward recurrence times , future discounted reward process , IMRL and DFR distributions , monotonicity properties for stochastic processes , renewal theory

Rights: Copyright © 1980 Institute of Mathematical Statistics


Vol.8 • No. 2 • April, 1980
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