## The Annals of Probability

### Further Monotonicity Properties for Specialized Renewal Processes

Mark Brown

#### Abstract

Define $Z(t)$ to be the forward recurrence time at $t$ for a renewal process with interarrival time distribution, $F$, which is assumed to be IMRL (increasing mean residual life). It is shown that $E\phi(Z(t))$ is increasing in $t \geq 0$ for all increasing convex $\phi$. An example demonstrates that $Z(t)$ is not necessarily stochastically increasing nor is the renewal function necessarily concave. Both of these properties are known to hold for $F$ DFR (decreasing failure rate).

#### Article information

Source
Ann. Probab. Volume 9, Number 5 (1981), 891-895.

Dates
First available in Project Euclid: 19 April 2007

http://projecteuclid.org/euclid.aop/1176994317

Digital Object Identifier
doi:10.1214/aop/1176994317

Mathematical Reviews number (MathSciNet)
MR628882

Zentralblatt MATH identifier
0489.60089

JSTOR