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On the Recognition and Structure of Probability Generating Functions

  • Anthony G. Pakes
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 84)

Abstract

If
$$ M\left( s \right) = 1 - {e^{ - \pi (s)}} $$
is a probability generating function, the coefficients π j in the MacLaurin expansion π(s) comprise a harmonic renewal sequence. A simple sufficient condition is given which ensures that a non-negative sequence is harmonic renewal. This condition covers the case of the limiting conditional law of a subcritical Markov branching process.

Examples are given illustrating the limitation of the criterion. The parallel problem for continuous laws and its relation to the CB-process is discussed.

Key words.

Generating functions Harmonic renewal functions Infinitely divisible laws Branching processes 

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References

  1. [1]
    Asmussen, S. Hering, H. (1983) Branching Processes. Birkhäuser, Boston.zbMATHGoogle Scholar
  2. [2]
    Bingham, N.H. (1976) Continuous branching processes and spectral positivity. Stoch. Processes Appl. 4, 217–142.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Bingham, N.H., Goldie, C.M. Teugels, J.L. (1987) Regular Variation. C.U.P., Cambridge.Google Scholar
  4. [4]
    Embrechts, P. & Omey, E. (1984) Functions of power series. Yokohama Math. J. 32,77–88.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. 2, 2nd ed. Wiley, New York.Google Scholar
  6. [6]
    Greenwood, P., Omey, E. & Teugels, J.L. (1982) Harmonic renewal functions. Z. Wahrscheinlichkeitsth. 59, 391–409.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Katti, S.K. (1967) Infinite divisibility of integer valued random variables. Ann. Math. Statist. 38, 1306–1308.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Kingman, J.F.C. (1972) Regenerative Phenomena. Wiley, London.zbMATHGoogle Scholar
  9. [9]
    Lukacs, E. (1970) Characteristic Functions, 2nd ed. Griffin, London.Google Scholar
  10. [10]
    Moran, P.A.P. (1968) An Introduction to Probability Theory. Clarendon Press, Oxford.zbMATHGoogle Scholar
  11. [11]
    Pakes, A.G. (1994) The subcritical CB-process and F—stable laws. In preparation.Google Scholar
  12. [12]
    Pakes, A.G. Speed, T.P. (1977) Lagrange distributions and their limit theorems. SIAM J. Appl. Math. 32, 745–754.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    Pakes, A.G. Trajtsman, A.C. (1985) Some properties of continuous-statebranching processes with applications to Bartozyuiski’s virus model. Adv. Appl. Prob. 17,23–41.zbMATHCrossRefGoogle Scholar
  14. [14]
    Port, S.C. (1963) An elementary probability approach to fluctuation theory. J. Math. Anal. Appl. 6, 109–151.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Spitzer, F. (1976) Principles of Random Walk 2nd ed. Springer Verlag, New York.CrossRefGoogle Scholar
  16. [16]
    Stromberg, K.R. (1981) Introduction to Classical Real Analysis. Wadsworth, Belmont, CA.Google Scholar
  17. [17]
    Yang, Y.S. (1973) Asymptotic properties of the stationary measure of a Markov branching process. J.Appl. Pro b. 10, 447–450.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Anthony G. Pakes
    • 1
  1. 1.Department of MathematicsUniversity of Western AustraliaNedlandsAustralia

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