Frontiers of Mathematics in China >

2014 , Vol. 9 >Issue 1: 63 - 79

DOI: https://doi.org/10.1007/s11464-013-0226-9

RESEARCH ARTICLE

Limit theorems for flows of branching processes

  • Hui HE 1 ,
  • Rugang MA , 2
Expand
  • 1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
  • 2. School of Applied Mathematics, Central University of Finance and Economics, Beijing 100081, China

Received date: 31 Mar 2012

Accepted date: 31 May 2012

Published date: 01 Feb 2014

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg
Fold

Abstract

We construct two kinds of stochastic flows of discrete Galton-Watson branching processes. Some scaling limit theorems for the flows are proved, which lead to local and nonlocal branching superprocesses over the positive half line.

Key words： Stochastic flow; Galton-Watson branching process; continuousstate branching process; superprocess; nonlocal branching

Cite this article

Hui HE , Rugang MA . Limit theorems for flows of branching processes[J]. Frontiers of Mathematics in China, 2014 , 9(1) : 63 -79 . DOI: 10.1007/s11464-013-0226-9

References

Publishing order | Descend order by publishing year | Descend order by cited within
1
Abraham R, Delmas J F. A continuum tree-valued Markov process. Ann Probab, 2012, 40: 1167-1211

DOI

2
Aldous D, Pitman J. Tree-valuedMarkov chains derived from Galton-Watson processes. Ann Inst Henri Poincaré Probab Stat, 1998, 34: 637-686

DOI

3
Bakhtin Y. SPDE approximation for random trees. Markov Process Related Fields, 2011, 17: 1-36

DOI

4
Bertoin J, Le Gall J F. Stochastic flows associated to coalescent processes. Probab Theory Related Fields, 2003, 126: 261-288

DOI

5
Bertoin J, Le Gall J F. Stochastic flows associated to coalescent processes II: Stochastic differential equations. Ann Inst Henri Poincaré Probab Stat, 2005, 41: 307-333

DOI

6
Bertoin J, Le Gall J F. Stochastic flows associated to coalescent processes III: Limit theorems. Illinois J Math, 2006, 50: 147-181

7
Dawson D A, Li Z. Skew convolution semigroups and affine Markov processes. Ann Probab, 2006, 34: 1103-1142

DOI

8
Dawson D A, Li Z. Stochastic equations, flows and measure-valued processes. Ann Probab, 2012, 40: 813-857

DOI

9
Ethier S N, Kurtz T G. Markov Processes: Characterization and Convergence. New York: Wiley, 1986

DOI

10
Fu Z, Li Z. Stochastic equations of non-negative processes with jumps. Stochastic Process Appl, 120: 306-330

DOI

11
Hewitt E, Stromberg K. Real and Abstract Analysis. Berlin: Springer, 1975

12
Jiřina M. Stochastic branching processes with continuous state space. Czechoslovak Math J, 1958, 8: 292-313

13
Kawazu K, Watanabe S. Branching processes with immigration and related limit theorems. Theory Probab Appl, 1971, 16: 36-54

DOI

14
Lamperti J. The limit of a sequence of branching processes. Z Wahrsch verw Geb, 1967, 7: 271-288

15
Li Z. A limit theorem for discrete Galton-Watson branching processes with immigration. J Appl Probab, 2006, 43: 289-295

DOI

16
Li Z. Measure-Valued Branching Markov Processes. Berlin: Springer, 2011

DOI

17
Li Z. Path-valued branching processes and nonlocal branching superprocesses. Ann Probab (to appear)

18
Li Z, Ma C. Catalytic discrete state branching models and related limit theorems. J Theoret Probab, 2008, 21: 936-965

DOI

19
Roelly S. A criterion of convergence of measure-valued processes: Application to measure branching processes. Stochastics, 1986, 17: 43-65

DOI

Options
Outlines

/

〈 〉