Advertisement

Renewal Processes

  • Kosto V. Mitov
  • Edward Omey
Chapter
Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Abstract

In this chapter, we give an overview of ordinary renewal theory on the positive half-line. We state the basic definitions and prove the important theorems on the renewal process and the renewal function. We give a detailed proof of the elementary renewal theorem and Blackwell’s theorem and we also provide some rate of convergence results. Moreover, we discuss limit theorems for the lifetime processes and give a short overview of delayed renewal processes.

Keywords

Ordinary renewal process Delayed renewal process Elementary renewal theorem Blackwell’s theorem Key renewal theorem Lifetime processes Rate of convergence Stationary renewal process 

References

  1. 1.
    Alsmeyer, G.: Erneuerungstheorie. Teubner, Stuttgart (1991)CrossRefzbMATHGoogle Scholar
  2. 2.
    Asmussen, S.: Applied Probability and Queues, 2nd edn. Springer, New York (2003)zbMATHGoogle Scholar
  3. 3.
    Choquet, G., Deny, J.: Sur l’equation de convolution \(\mu =\mu *\sigma \). C. R. Acad. Sci. Paris 250, 799–801 (1960)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Daley, D.J., Ver-Jones, D.: An Introduction to the Theory of Point Processes, vol. 1, 2nd edn. Springer, Berlin (2003)Google Scholar
  5. 5.
    Dippon, J., Walk, H.: Simplified analytical proof of Blackwell’s renewal theorem. Stat. Probab. Lett. 74, 15–20 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Feller, W.: An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York (1971)zbMATHGoogle Scholar
  7. 7.
    Kac, M.: A remark on Wienner’s Tauberian theorem. Proc. Am. Math. Soc. 16, 1155–1157 (1965)zbMATHGoogle Scholar
  8. 8.
    Lindvall, T.: A probabilistic proof of Blackwell’s renewal theorem. Ann. Probab. 5(3), 482–485 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Ney, P.: A refinement of the coupling method in renewal theory. Stoch. Proc. Appl. 11, 11–26 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Omey, E., Vesilo, R.: Local limit theorems for shock models. HUB Research paper 2011/23. HUB, Brussel (2011)Google Scholar
  11. 11.
    Serfozo, R.: Basic of Applied Stochastic Processes. Springer, New York (2009)CrossRefGoogle Scholar
  12. 12.
    Smith, W.L.: Asymptotic renewal theorems. Proc. R. Soc. Edinb. Sect. A 64, 9–48 (1954)Google Scholar

Copyright information

© The Author(s) 2014

Authors and Affiliations

  1. 1.Aviation FacultyNational Military University “Vasil Levski”Dolna MitropoliaBulgaria
  2. 2.Faculty of Economics and BusinessKU LeuvenBrusselsBelgium

Personalised recommendations