Covariance between the forward recurrence time and the number of renewals
Volume 9, Issue 1 (2022), pp. 1–16
Pub. online: 15 December 2021
Type: Research Article
Open Access
Open Access
Received
13 August 2021
13 August 2021
Revised
20 November 2021
20 November 2021
Accepted
26 November 2021
26 November 2021
Published
15 December 2021
15 December 2021
Notes
In memory of my beloved father Ioannis Losidis
Abstract
Recurrence times and the number of renewals in $(0,t]$ are fundamental quantities in renewal theory. Firstly, it is proved that the upper orthant order for the pair of the forward and backward recurrence times may result in NWUC (NBUC) interarrivals. It is also demonstrated that, under DFR interarrival times, the backward recurrence time is smaller than the forward recurrence time in the hazard rate order. Lastly, the sign of the covariance between the forward recurrence time and the number of renewals in $(0,t]$ at a fixed time point t and when $t\to \infty $ is studied assuming that the interarrival distribution belongs to certain ageing classes.
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