The Annals of Probability

Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes

Mark Brown

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Abstract

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 $t \geqslant 0$.

Article information

Source
Ann. Probab. Volume 8, Number 2 (1980), 227-240.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
http://projecteuclid.org/euclid.aop/1176994773

JSTOR
links.jstor.org

Digital Object Identifier
doi:10.1214/aop/1176994773

Mathematical Reviews number (MathSciNet)
MR566590

Zentralblatt MATH identifier
0429.60084

Subjects
Primary: 60K05: Renewal theory
Secondary: 60699

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

Citation

Brown, Mark. Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes. Ann. Probab. 8 (1980), no. 2, 227--240. doi:10.1214/aop/1176994773. http://projecteuclid.org/euclid.aop/1176994773.


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