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Improved bounds for the solutions of renewal equations

Published online by Cambridge University Press:  06 June 2022

Stathis Chadjiconstantinidis
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80, M. Karaoli & A. Dimitriou St., 18534 Piraeus, Greece. E-mail: stch@unipi.gr
George Tzavelas
Affiliation:
Department of Statistics and Insurance Science, University of Piraeus, 80, M. Karaoli & A. Dimitriou St., 18534 Piraeus, Greece. E-mail: stch@unipi.gr

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.

Type
Research Article
Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press

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