Skip to main content
×
×
Home

Functional equations and the Galton-Watson process

  • E. Seneta (a1)
Extract

In the present exposition we are concerned only with the simple Galton-Watson process, initiated by a single ancestor (Harris (1963), Chapter I). Let denote the probability generating function of the offspring distribution of a single individual. Our fundamental assumption, which holds throughout the sequel, is that fj ≠ 1, j = 0,1,2, …; in particular circumstances it shall be necessary to strengthen this to 0 < f 0F(0) < 1, which is the relevant assumption when extinction behaviour is to be considered. (Even so, our assumptions will always differ slightly from those of Harris (1963), p. 5.)

Copyright
References
Hide All
Bartlett, M. S. (1955) An Introduction to Stochastic Processes. Cambridge University Press, Cambridge.
Breny, H. (1962) Sur un point de la théorie des files d'attente. Ann. Soc. Sci. Bruxelles 76, 512.
Coifman, R. (1965) Sur l'unicité des solutions de l'équation d'Abel-Schröder et l'itéation continue. J. Aust. Math. Soc. 5, 3647.
Dunne, M. C. and Potts, R. B. (1965) Analysis of a computer control of an isolated intersection. Proceedings of the Third International Symposium on the Theory of Traffic Flow, New York.
Feller, W. (1957) An Introduction to Probability Theory and its Applications. Vol. I. Wiley, New York.
Gnedenko, B. V. and Kovalenko, I. N. (1966) Introduction to the Theory of Mass Service (in Russian). “Nauka”, Moscow.
Haldane, J. B. S. (1949) Some statistical problems arising in genetics. J. R. Statist. Soc. B 11, 114.
Harris, T. E. (1963) The Theory of Branching Processes. Springer, Berlin.
Heathcote, C. R. (1965) A branching process allowing immigration. J. R. Statist. Soc. B 27, 138143.
Heathcote, C. R. (1966) Corrections and comments on the paper “A branching process allowing immigration”. J. R. Statist. Soc. B 28, 213217.
Heathcote, C. R., Seneta, E. and Vere-Jones, D. (1967) A refinement of two theorems in the theory of branching processes. Teor. Veroyat. Primen. 12, 341346.
Joffe, A. (1967) On the Galton-Watson branching process with mean less than one. Ann. Math. Statist. 38, 264266.
Karlin, S. and McGregor, J. L. (1966) Spectral theory of branching processes. I. Z. Wahrscheinlichkeitsth. 5, 633.
Kendall, D. G. (1966a) Branching processes since 1873. J. London Math. Soc. 41, 385406.
Kendall, D. G. (1966b) On supercritical branching processes with a positive chance of extinction. Research Papers in Statistics. Festschrift for J. Neyman, 157165.
Kesten, H. (1966) (Private communication).
Kesten, H., Ney, P. and Spitzer, F. (1966) The Galton-Watson process with mean one and finite variance. Teor. Veroyat. Primen. 11, 579611.
Kesten, H. and Stigum, B. P. (1966) A limit theorem for multidimensional Galton-Watson processes. Ann. Math. Statist. 37, 12111223.
Khintchine, A. (1948) Asymptotische Gesetze der Wahrscheinlichkeitsrechnung. Chelsea, New York.
Kingman, J. F. C. (1961) The single server queue in heavy traffic. Proc. Camb. Phil. Soc. 57, 902904.
Kneser, H. (1950) Reelle analytische Lösungen der Gleichung ϕ(ϕ)(x)) = e x und ver-wandter Funktionalgleichungen. J. reine angew. Math. 187, 5667.
Kolmogorov, A. N. (1938) Zur Lösung einer biologischen Aufgabe. Comm. Math. Mech. Chebychev Univ., Tomsk 2, 16.
Kuczma, M. (1960) Remarques sur quelques théorèmes de J. Anastassiadis. Bull. Sci. Math. 84, 98102.
Kuczma, M. (1961) Sur une équation fonctionelle. Mathematica (Cluj) 3, 7987.
Kuczma, M. (1963) On the Schröder equation. Rozprawy Mat. 34.
Kuczma, M. (1964a) A survey of the theory of functional equations. Publikacije Elektrotechničkog Fakulteta, Univerzitet u Beogradu, Serija: Matematika i Fizika, No. 130.
Kuczma, M. (1964b) Note on Schröder's functional equation. J. Aust. Math. Soc. 4, 149151.
Kuczma, M. (1965) On convex solutions of Abel's functional equation. Bull. Acad. Polon. Sci. (Math., Astr., Phys.) 13, 645648.
Kuczma, M. (1966) Sur l'équation de Böttcher. Mathematica (Cluj) 8, 279285.
Kuczma, M. (1967) Un théorème de l'unicité pour l'équation fonctionelle de Böttcher. Mathematica (Cluj) (to appear).
Kuczma, M. (1968) Functional Equations in a Single Variable. Monografie Matematyczne Vol. 46, Warsaw.
Kuczma, M. and Smajdor, A. (1967) Note on iteration of concave functions. Amer. Math. Monthly 74, 401402.
Levinson, N. (1959) Limiting theorems for Galton-Watson branching process. Illinois J. Math. 3, 554565.
Levy, P. (1928) Fonctions à croissance régulière et itération d'ordre fractionnaire. Ann. Mat. Pura Applic. 5, 269298.
Nagaev, A. V. (1961) A refinement of certain theorems of the theory of branching random processes (in Russian). Trudy Tashkent. Univ. (im Lenin) 189, 5563.
Nagaev, A. V. and Badalbaev, I. (1967) A refinement of certain theorems on branching random processes (in Russian). Litovsk. Mat. Sb. 7, 129136.
Papangelou, F. (1967) A lemma on the Galton-Watson process. Z. Wahrscheinlichkeitsth. (to appear).
Picard, E. (1950) Leçons sur Quelques Équations Fonctionelles. Gauthier-Villars, Paris.
Reuter, G. E. H. (1968) (Private communication).
Seneta, E. (1967) The Galton-Watson process with mean one. J. Appl. Prob. 4, 489495.
Seneta, E. (1968a) Topics in the Theory and Applications of Markov Chains. , Australian National University.
Seneta, E. (1968b) The stationary distribution of a branching process allowing immigration: a remark on the critical case. J. R. Statist. Soc. B 30, 176179.
Seneta, E. (1968c) On asymptotic properties of subcritical branching processes. J. Aust. Math. Soc. 8, 671682.
Seneta, E. (1969) On Koenigs' ratios for iterates of real functions (to appear).
Seneta, E. and Vere-Jones, D. (1966) On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Prob. 3, 403434.
Skellam, J. G. (1949) The probability distribution of gene-differences in relation to selection, mutation, and random extinction. Proc. Camb. Phil. Soc. 45, 364367.
Slack, R. S. (1968) A branching process with mean one and possibly infinite variance. Z. Wahrscheinlichkeitsth. 9, 139145.
Stigum, B. P. (1966) A theorem on the Galton-Watson process. Ann. Math. Statist. 37, 695698.
Szekeres, G. (1958) Regular iteration of real and complex functions. Acta Math. 100, 203258.
Szekeres, G. (1960) On a theorem of Paul Lévy. Magyar Tud. Akad. Mat. Kutato Int. Közl. A 5, 277282.
Vere-Jones, D. (1966) Simple stochastic models for the release of quanta of transmitter from a nerve terminal. Aust. J. Statist. 8, 5363.
Yaglom, A. M. (1947) Certain limit theorems in the theory of branching stochastic processes. (In Russian.) Doklady Akad. Nauk S.S.S.R. n.s., 56, 795798.
Zolotarev, V. M. (1957) A refinement of several theorems in the theory of branching stochastic processes. (in Russian.) Teor. Veroyat. Primen. 2, 256266.
Karamata, J. (1953) Über das asymptotische Verhalten der Folgen die durch Iteration definiert sind. (In Serbian; German summary.) Recueil des Travaux de l'Académie Serbe des Sciences XXXV, Institut Mathématique No. 3, 4560.
Quine, M. P. and Seneta, E. (1969) A limit theorem tor the Galton-Watson process with immigration. Aust. J. Statist. 11, (to appear).
Seneta, E. (1968) On recent theorems concerning the supercritical Galton-Watson process. Ann. Math. Statist. 39, 20982102.
Recommend this journal

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

Advances in Applied Probability
  • ISSN: 0001-8678
  • EISSN: 1475-6064
  • URL: /core/journals/advances-in-applied-probability
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: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed