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  • Improved bounds for the solutions of renewal equations

  • Stathis Chadjiconstantinidis, George Tzavelas
  • Journal:
    Probability in the Engineering and Informational Sciences / Volume 37 / Issue 3 / July 2023
    Published online by Cambridge University Press:
    06 June 2022, pp. 740-777
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  • 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.