Skip to main content
SpringerLink
Log in

On the Integral Equation of Renewal Theory

  • Chapter
  • First Online:
Mathematical Demography

Part of the book series: Demographic Research Monographs ((DEMOGRAPHIC))

  • 1586 Accesses

Abstract

Feller’s paper is a rigorous treatment of renewal theory, and to assist the reader his principal results are summarized below in demographic form and notation.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 139.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 179.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 179.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

  1. 1.

    Note: Lotka’s paper [8] contains a list of 74 papers on the subject published before 1939. The following list is to bring Lotka's list up to June 1941; however no claims to completeness are made.

References

I. Papers on the integral equation of renewal theory

  1. A. W. Brown, A note on the use of a Pearson type III function in renewal theory,” Annals of Math. Stat. Vol. 11 (1940), pp. 448–453.

    Article  Google Scholar 

  2. H. Hadwiger, “Zur Frage des Beharrungszustandes bei kontinuierlich sich erneuernden Gesamtheiten,” Archiv f. mathem. Wirtschafts- und Sozialforschung, Vol. 5 (1939), pp. 32–34.

    Google Scholar 

  3. H. Hadwiger, “Über die Integralgleichung der Bevölkerungstheorie,” Mitteilungen Verein. schweizer Yersicherungsmathematiker (Bull. Assoc. Actuaires suisses), Vol. 38 (1939), pp. 1–14.

    Google Scholar 

  4. H. Hadwiger, “Eine analytische Reproduktionsfunktion für biologische Gesamtheiten,” Skand. Aktuarietidskrift (1940), pp. 101–113.

    Google Scholar 

  5. H. Hadwiger, “Natürliche Ausscheidefunktionen für Gesamtheiten und die Lösung der Erneuerungsgleichung,” Mitteilungen Verein. Schweiz. Versich.-Math., Vol. 40 (1940), pp. 31–39.

    Google Scholar 

  6. H. Hadwiger and W. Ruchti, “Über eine spezielle Klasse analytischer Geburtenfunktionen,” Metron, Vol. 13 (1939), No.4, pp.17–26.

    Google Scholar 

  7. A. Linder, “Die Vermehrungsrate der stabilen Bevölkerung,” Archiv f. mathem. Wirtschafts- und Sozialforschung, Vol. 4 (1938), pp. 136–156.

    Google Scholar 

  8. A. Lotka, “A contribution to the theory of self-renewing aggregates, with special reference to industrial replacement,” Annals of Math. Stat., Vol. 10 (1939), pp. 1–25.

    Article  Google Scholar 

  9. A. Lotka, “On an integral equation in population analysis,” Annals of Math. Stat., Vol. 10 (1939), pp. 144–161.

    Article  Google Scholar 

  10. A. Lotka, “Théorie analytique des associations biologiques II,” Actualités Scientifiques No. 780, Paris, 1939.

    Google Scholar 

  11. A. Lotka, “The theory of industrial replacement,” Skand. Aktuarietidskrift (1940), pp. 1–14.

    Google Scholar 

  12. A. Lotka, “Sur une équation intégrale de l’analyse démographique et industrielle,” Mitt. Yerein. Schweiz. Yersich.-Math., Vol. 40 (1940), pp. 1–16.

    Google Scholar 

  13. H. Münzner, “Die Erneuerung von Gesamtheiten,” Archiv f. math. Wirtschafts- u. Sozialforschung, Vol. 4 (1938).

    Google Scholar 

  14. G. A. D. Preinreich, “The theory of industrial replacement,” Skand. Aktuarietidskrift (1939), pp. 1–19.

    Google Scholar 

  15. E. C. Rhodes, “Population mathematics, I, II, III,” Roy. Stat. Soc. Jour., Vol. 103 (1940),pp.61–89,218–245,362–387.

    Google Scholar 

  16. H. Richter, “Die Konvergenz der Erneuerungsfunktion,” Blätter f. Versicherungamathcmatik, Vol. 5 (1940), pp. 21–35.

    Google Scholar 

  17. H. Hadwiger, “Über eine Funktionalgleichung der Bevölkerungstheorie und eine spezielle Klassc analytischer Lösungen,” Bl. f. Versicherungsmathematik, Vol. 5 (1941), pp. 181–188.

    Google Scholar 

  18. G. A. D. Prienreich, “The present status of renewal theory,” Waverly Press, Baltimore (1940).

    Google Scholar 

  19. R. V. Churchill, “The inversion of the Laplace transformation by a direct expansion in series and its application to boundary-value problems,” Math. Zeits., Vol. 42 (1937), pp. 567–579.

    Article  Google Scholar 

  20. G. Doetsch, Theorit und Anwendung der Laplace Transformation. J. Springer, Berlin, 1937.

    Google Scholar 

  21. W. Feller, “Completely monotone functions and sequences,” Duke Math. Jour., Vol. 5 (1939), pp. 661–674.

    Article  Google Scholar 

  22. A. Haar, “Über asymptotische Entwicklungen von Funktionen,” Math. Ann., Vol. 96 (1927), pp. 69–107.

    Article  Google Scholar 

  23. R. E. A. C. Paley and N. Wiener, “Notes on the theory and application of Fourier transforms, VII. On the Volterra equation,” Amer. Math. Soc. Trans., Vol. 35 (1933), pp. 785–791.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Harvard University, Cambridge, USA

    David P. Smith

  2. Department of Sociology, Harvard University, Cambridge, USA

    Nathan Keyfitz

Authors
  1. David P. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nathan Keyfitz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of California, Berkeley, Berkeley, California, USA

    Kenneth W. Wachter

  2. School for Advanced Studies in the Social Sciences, Paris, France

    Hervé Le Bras

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Smith, D.P., Keyfitz, N. (2013). On the Integral Equation of Renewal Theory. In: Wachter, K., Le Bras, H. (eds) Mathematical Demography. Demographic Research Monographs. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-35858-6_16

Download citation

Publish with us

Policies and ethics

Search

Navigation