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Improved bounds for the solutions of renewal equations

Published online by Cambridge University Press:  06 June 2022

Stathis Chadjiconstantinidis Open the ORCID record for Stathis Chadjiconstantinidis [Opens in a new window]  and
George Tzavelas
Stathis Chadjiconstantinidis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80, M. Karaoli & A. Dimitriou St., 18534 Piraeus, Greece. E-mail: [email protected]
George Tzavelas
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80, M. Karaoli & A. Dimitriou St., 18534 Piraeus, Greece. E-mail: [email protected]
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Abstract

Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.

Keywords

Forward recurrence timeIMRL (DMRL) classNBUE (NWUE) classRenewal densityRenewal functionRenewal type equation
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 37 , Issue 3 , July 2023 , pp. 740 - 777
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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