×

zbMATH — the first resource for mathematics

A criterion of convergence of measure-valued processes: Application to measure branching processes. (English) Zbl 0598.60088
Author’s summary: In this paper martingale properties of a measure branching process are investigated. Uniqueness and continuity of this process are proven by a martingale approach. For the existence, the author approximates the measure branching process by a sequence of infinite particle branching diffusion processes, and shows the convergence in distribution by a new criterion for measure-valued processes. He also gives properties about local structure of the process.
Reviewer: D.Dawson

MSC:
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G57 Random measures
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aldous D., Ann. Prob. 6 (1978)
[2] Athreya K. J., Branching Processes (1972) · Zbl 0259.60002
[3] Billingsley P., Convergence of Probability Measures (1968) · Zbl 0172.21201
[4] Blumenthal R. M., Markov Processes and Potential Theory (1968) · Zbl 0169.49204
[5] Cuppens R., Decomposition of Multivariate Probability (1975) · Zbl 0363.60012
[6] DOI: 10.1016/0047-259X(75)90054-8 · Zbl 0299.60050 · doi:10.1016/0047-259X(75)90054-8
[7] DOI: 10.1007/BF00532877 · Zbl 0343.60001 · doi:10.1007/BF00532877
[8] Dawson D. A., Col. Math. Soc. J. Bolyai 40 (1978)
[9] Dawson D. A., Advances in Probability 5 (1978)
[10] DOI: 10.1214/aop/1176994991 · Zbl 0411.60084 · doi:10.1214/aop/1176994991
[11] El Karoui N., Non linear evolution equations and dfunctionals of measure valued branching processes · Zbl 0587.60083
[12] El Karoui N., Study of a general class of measure-valued branching processes with stochastic calculus
[13] Fouque J. P.., La convergence en loi pour les processes a valeurs dans un espace nucleaire · Zbl 0545.60012
[14] Guelfand I. M., Les Distributions 4 (1967) · Zbl 0115.10102
[15] Iscoe I., A weighted occupation time for a class of measure-valued branching processes · Zbl 0555.60034 · doi:10.1007/BF00366274
[16] Jacod J., Proc. Durham Symp. (1980)
[17] Jirina M., Czechoslovak Math. J. 8 (1958)
[18] Kallenberg O., Random Measures (1976)
[19] DOI: 10.1137/1116003 · doi:10.1137/1116003
[20] Kurtz T. G., Approximation of Population Processes 16 (1981) · Zbl 0465.60078
[21] Lamperti J., Bull. Amer. Math. Soc. 136 (1967)
[22] Matthes K., Infinitely Divisible Point Processes (1979) · Zbl 0267.60060
[23] Pazy A., Semigroups of Linear Operators: Applications to Partial Differential Equations, Applied Math. 44 (1983) · Zbl 0516.47023
[24] Rebolledo R., C.R. Acad. Sci., Paris 290 (1980)
[25] Roelly- Coppoletta S., These Univ. Paris 6 (1984)
[26] DOI: 10.1007/BF00736006 · Zbl 0327.60047 · doi:10.1007/BF00736006
[27] Watanabe S., H. Math. Kyoto Univ. 8 (1968)
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.