×

zbMATH — the first resource for mathematics

Convergence of critical multitype Galton-Watson branching processes. (English) Zbl 0608.60080
Let \(Z_ n\) denote a positive regular, non-singular, d-type, critical Galton-Watson process with not necessarily finite second moments of its offspring probability generating function F(s). Define \(a_ n=v(1-F_ n(0))\), when \(F_ n\) denotes the n-th iterate of F and v the normalized left eigenvector corresponding to the (maximal) eigenvalue 1 of the matrix of the offspring expectations. Let u denote the corresponding right eigenvector.
Considering \(Z_{[nt]}\) as a random element in the appropriate function space the author can prove two diffusion type limit theorems by increasing the initial population size if all the second moments of \(Z_ 1\) are finite: Firstly, not conditioning on non-extinction for \(\hat Y_ n(t)=a_ nZ_{[nt]}u\) with \(\hat Y_ n(0)=x_ 0+o(1)\) and, secondly, conditioning on non-extincting for \(Y_ n(t)=a_ nZ_{[nt]}u| Z_ n\neq 0\) with \(Z_ 0=a_ n^{-1}x_ 0v+o(a_ n^{-1}).\)
This generalizes results of T. Lindvall [J. Appl. Probab. 9, 445- 450 (1972; Zbl 0238.60063), J. Lamperti and P. Ney [Teor. Veroyatn. Primen. 13, 126-137 (1968; Zbl 0253.60073)] and T. Nakagawa [J. Multivariate Anal. 12, 161-177 (1982; Zbl 0492.60081)] and is obtained by convergence theorems for the finite-dimensional distributions under Vatutin’s infinite variance condition.
Reviewer: L.Edler

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F05 Central limit and other weak theorems
60J60 Diffusion processes
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Billingsley, P., Convergence of probability measures, (1968), John Wiley New York · Zbl 0172.21201
[2] Chistyakov, V.P., On convergence of branching processes to diffusion processes, Theor. probability appl., 15, 707-710, (1970) · Zbl 0232.60068
[3] Esty, W.W., Critical age-dependent branching process diffusions, Stoch. proc. appl., 3, 209-220, (1975) · Zbl 0312.60042
[4] Feller, W., Diffusion processes in genetics, (), 227-256
[5] Goldstein, M.I.; Hoppe, F.M., Necessary conditions for normed convergence of critical multitype bienaymé-Galton-Watson processes without variance, J. multivariate anal., 9, 55-62, (1978) · Zbl 0382.60092
[6] Harris, T.E., The theory of branching processes, (1963), Springer Berlin · Zbl 0117.13002
[7] Joffe, A.; Spitzer, F., On multitype branching processes with ϱ ⩽ 1, J. math. anal. appl., 19, 409-430, (1967) · Zbl 0178.19504
[8] Lamperti, J.; Ney, P., Conditioned branching processes and their limiting diffusions, Theor. probability appl., 13, 128-139, (1968) · Zbl 0253.60073
[9] Lindvall, T., Convergence of critical Galton-Watson branching processes, J. appl. prob., 9, 445-450, (1972) · Zbl 0238.60063
[10] Lindvall, T., Weak convergence of probability measures and random functions in the function space D[0, ∞], J. appl. prob., 10, 109-121, (1973) · Zbl 0258.60008
[11] Nakagawa, T., Reverse processes and some limit theorems of multitype Galton-Watson processes, J. multivariate anal., 12, 161-177, (1982) · Zbl 0492.60081
[12] Stroock, D.W.; Varadhan, S.R.S., Multidimensional diffusion processes, (1979), Springer Berlin
[13] Vatutin, V.A., Limit theorems for critical multitype Markov branching processes with infinite second moments, Matemacheskii sbornik, 103, 253-264, (1977), (in Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.