Skip to main content
×
Home
    • Aa
    • Aa

SOME NEW BOUNDS FOR THE RENEWAL FUNCTION

  • Konstadinos Politis (a1) and Markos V. Koutras (a1)
Abstract

In the literature, most of the bounds for the renewal function U(x) corresponding to a lifetime distribution F are given in terms of the first two moments of F only. The best general upper bound of this type is the one given in Lorden (1970). In the present article, we show that improved bounds can be obtained if one exploits the specific form of the distribution F. We derive a bound that improves upon Lorden's, at least on an interval [0,a) with a ≤ ∞, and we give both sufficient and necessary conditions for this improvement to hold uniformly for x ≥ 0. Refined upper as well as lower bounds are given for the case where F belongs to a class of distributions with monotone aging or when the renewal density is monotone.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

Brown, M. (1980). Bounds, inequalities, and monotonicity properties for some specialised renewal processes. Annals of Probability 8: 227240.

Cai, J. & Garrido, J. (1999). A unified approach to the study of tail probabilities of compound distributions. Journal of Applied Probability 36(4): 10581073.

Chen, Y-H. (1994). Classes of life distributions and renewal counting processes. Journal of Applied Probability 31: 11101115.

Daley, D.J. (1980). Tight bounds for the renewal function of a random walk. Annals of Probability 8: 615621.

Lin, X.S. (1996). Tail of compound distributions and excess time. Journal of Applied Probability 33: 184195.

Lorden, G. (1970). On excess over the boundary. Annals of Mathematical Statistics 41: 520527.

Marshall, K.T. (1973). Linear bounds on the renewal function. SIAM Journal of Applied Mathematics 24: 245250.

Ross, S. (2003). A note on the insurance risk problem. Probability in the Engineering and Informational Sciences 17: 199203.

Sengupta, D. & Nanda, A.K. (1999). Log-concave and concave distributions in reliability. Naval Research Logistics 46: 419433.

Shaked, M. & Zhu, H. (1992). Some results on block replacement policies and renewal theory. Journal of Applied Probability 29: 932946.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Probability in the Engineering and Informational Sciences
  • ISSN: 0269-9648
  • EISSN: 1469-8951
  • URL: /core/journals/probability-in-the-engineering-and-informational-sciences
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 1
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 47 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th May 2017. This data will be updated every 24 hours.