The renewal-process stationary-excess operator

1 March 1985

journal article
research article
Published by Cambridge University Press (CUP) in Journal of Applied Probability

Vol. 22 (01) , 156-167
https://doi.org/10.1017/s0021900200029089

Abstract

This paper describes the operator mapping a renewal-interval distribution into its associated stationary-excess distribution. This operator is monotone for some kinds of stochastic order, but not for the usual stochastic order determined by the expected value of all non-decreasing functions. Conditions for a renewal-interval distribution to be larger or smaller than its associated stationary-excess distribution for several kinds of stochastic order are determined in terms of familiar notions of ageing. Convergence results are also obtained for successive iterates of the operator, which supplement Harkness and Shantaram (1969), (1972) and van Beek and Braat (1973).

Keywords

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