On the Integral Equation of Renewal Theory

  • William Feller
Part of the Biomathematics book series (BIOMATHEMATICS, volume 6)


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.


Characteristic Equation Monotone Function Parent Population Discontinuous Function Intrinsic Growth Rate 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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I. Papers on the integral equation of renewal theory

  1. [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.CrossRefzbMATHGoogle Scholar
  2. [2]
    H. Hadwiger, “Zur Frage des Beharrungszustandes bei kontinuierlich sich erneuernden Gesamtheiten,” Archiv f. mathem. Wirischaftsund Sozialforschung, Vol. 5 (1939), pp. 32–34.Google Scholar
  3. [3]
    H. Hadwiger, “Über die Integralgleichung der Bevölkerungstheorie,” Mitteilungen Verein, schweizer Versicherungsmathematiker (Bull. Assoc. Actuaires suisses), Vol. 38 (1939), pp. 1–14.zbMATHMathSciNetGoogle Scholar
  4. [4]
    H. Hadwiger, “Eine analytische Reproduktionsfunktion für biologische Gesamtheiten,” Skand. Aktuarietidskrift (1940), pp. 101–113.Google Scholar
  5. [5]
    H. Hadwiger, “Natürliche Ausscheidefunktionen für Gesamtheiten und die Lösung der Erneuerungsgleichung,” Mitteilungen Verein. Schweiz. Ver sich.-Math, Vol. 40 (1940), pp. 31–39.zbMATHMathSciNetGoogle Scholar
  6. [6]
    H. Hadwiger and W. Ruchti, “Über eine spezielle Klasse analytischer Geburtçn funktionen,” Metron, Vol. 13 (1939), No. 4, pp. 17–26.zbMATHMathSciNetGoogle Scholar
  7. [7]
    A. Linder, “Die Vermehrungsrate der stabilen Bevölkerung,” Archiv f. mathem. Wirtschaftsund Sozialforschung, Vol. 4 (1938), pp. 136–156.Google Scholar
  8. [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.CrossRefzbMATHGoogle Scholar
  9. [9]
    A. Lotka, “On an integral equation in population analysis,” Annals of Math. Stat., Vol. 10 (1939), pp. 144–161.CrossRefzbMATHGoogle Scholar
  10. [10]
    A. Lotka, “Théorie analytique des associations biologiques II,” Actualités Scienti fiques No. 780, Paris, 1939.Google Scholar
  11. [11]
    A. Lotka, “The theory of industrial replacement,” Skand. Aktuarietidskrift (1940), pp.1–14.Google Scholar
  12. [12]
    A. Lotka, “Sur une équation intégrale de l’analyse démographique et industrielle,” Mitt. Verein. Schweiz. Ver sich.-Math., Vol. 40 (1940), pp. 1–16.zbMATHMathSciNetGoogle Scholar
  13. [13]
    Ii. Münzner, “Die Erneuerung von Gesamtheiten,” Archiv f. math. Wirtschafts- u. Sozialforschung, Vol. 4 (1938).Google Scholar
  14. [14]
    G. A. D. Preinreich, “The theory of industrial replacement,” Skand. Aktuarietidskrift (1939), pp. 1–19.Google Scholar
  15. [15]
    EC. Rhodes, “Population mathematics, I, II, III,” Roy. Stat. Soc. Jour., Vol. 103 (1940), pp. 61–89, 218–245, 362–387.CrossRefGoogle Scholar
  16. [16]
    H. Richter, “Die Konvergenz der Erneuerungsfunktion,” Blätter f. Versicherungs mathematik, Vol. 5 (1940), pp. 21–35.Google Scholar
  17. [16a]
    H. HADWIGER, “Über eine Funktionalgleichung der Bevölkerungstheorie und eine spezielle Klasse analytischer Lösungen,” Bl. f. Versicherungsmathematik, Vol. 5 (1941), pp. 181–188.Google Scholar
  18. [16b]
    G. A. D. Prienreich, “The present status of renewal theory,” Waverly Press, Baltimore (1940).Google Scholar

II. Other papers quoted

  1. [17]
    E. 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.CrossRefMathSciNetGoogle Scholar
  2. [18]
    G. Doetsch, Theorie und Anwendung der Laplace Transformation. J. Springer, Berlin, 1937.Google Scholar
  3. [19]
    W. Feller, “Completely monotone functions and sequences,”Duke Math. Jour. Vol. 5 (1939), pp. 661–674.CrossRefMathSciNetGoogle Scholar
  4. [20]
    A. Haar, “über asymptotische Entwicklungen von Funktionen,” Math. Ann., Vol. 96 (1927), pp. 69–107.CrossRefMathSciNetGoogle Scholar
  5. [21]
    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

Copyright information

© Springer-Verlag Berlin · Heidelberg 1977

Authors and Affiliations

  • William Feller

There are no affiliations available

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