Abstract
It is shown that a 1-1 correspondence exists between the possible Yaglom conditional limits when a subcritical Galton-Watson process is initiated with an arbitrary probability distribution and the invariant measures of the process. This is proven by an examination of the relevant Schröder and Abel functional equations.
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References
Athreya, K. B. andNey, P. E.,Branching processes. Springer-Verlag, Berlin, 1972.
Heathcote, C. R., Seneta, E. andVere-Jones, D.,A refinement of two theorems in the theory of branching processes. Theor. Probability Appl.12 (1967), 297–301.
Hoppe, F. M.,Representations of invariant measures on multi-type Galton-Watson processes. Ann. Probability (1976)5 (1977), 291–297.
Hoppe, F. M.,On a result of Rubin and Vere-Jones concerning subcritical branching processes. J. Appl. Probability (1976)3 (1976), 804–806.
Rubin, H. andVere-Jones, D.,Domains of attraction for the subcritical Galton-Watson branching process. J. Appl. Probability5 (1968), 216–219.
Seneta, E.,On invariant measures for simple branching processes. J. Appl. Probability8 (1971), 43–51.
Seneta, E.,Regularly varying functions. Lecture Notes in Mathematics Vol. 508. Springer-Verlag, Berlin, 1976.
Seneta, E. andVere-Jones, D.,On quasi-stationary distributions in discrete-time Markov chains with a denumerable infinity of states. J. Appl. Probability3 (1966), 403–434.
Spitzer, F.,Two explicit Martin boundary constructions. InSymposium on Probability Methods in Analysis. Lecture Notes in Mathematics Vol. 31, pp. 296–298. Springer-Verlag, Berlin, 1967.
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Hoppe, F.M. On a Schröder equation arising in branching processes. Aeq. Math. 20, 33–37 (1980). https://doi.org/10.1007/BF02190491
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DOI: https://doi.org/10.1007/BF02190491