Refinement bounds for the expected number of renewal epochs over a finite interval

ABSTRACT

Primary quantities of interest in renewal theory are the expected number of renewals in $(0,t]$, known as renewal function (denoted by $U(t)$), and the expected number of renewals in $(t,t+h]$, given by the difference $U(t+h)-U(t)$.One of the most known bounds for the expected number of renewals over a finite interval is the one delivered by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The essential quantity of this bound is the difference $U(t+h)-U(t)-U(h)$. In this paper, initially, we offer bounds for the conditional tails of the backward and forward recurrence times, and we give a renewal-type equation for the difference $U(t+h)-U(t)-U(h)$. Employing those results, we refine the provided bound by Barlow and Proschan [Statistical theory of reliability and life testing. MD: Silver Spring; 1981. Reprint Edition. To begin with]. The results are illustrated with numerical examples.

KEYWORDS:

• Renewal process
• conditional probabilities
• forward recurrence time
• number of renewals
• variance of the number of renewals

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 60K05
• 60K10

Acknowledgments

I wish to thank both the Associate Editor and the referees for their helpful comments and suggestions which have improved considerably the clarity in the presentation of my results.

Disclosure statement

No potential conflict of interest was reported by the author(s).