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A Convergence Criterion for Measure-Valued Processes, and Application to Continuous Superprocesses

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Barcelona Seminar on Stochastic Analysis

Part of the book series: Progress in Probability ((PRPR,volume 32))

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Abstract

A criterion for weak convergence of measure-valued processes is proved, and it is exemplified by showing convergence of branching particle systems to continuous superprocesses.

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References

  1. Aldous, D. (1978). Stopping times and tightness, Ann. Probab. 6, 335–340.

    Article  Google Scholar 

  2. Aldous, D. (1989). Stopping times and tightness II, Ann. Probab. 17, 586–595.

    Article  Google Scholar 

  3. Bose, A. (1986). A law of large numbers for the scaled age distribution of linear birth-and-death processes, Canadian J. Stat. 14, 233–244.

    Article  Google Scholar 

  4. Cremers, H. and Kadelka, D. (1986). On weak convergence of integral functions of Stochastic processes with applications to processes taking paths in L E P, Stock. Proc. Appl. 21, 305–317.

    Article  Google Scholar 

  5. Dawson, D. A. (1991). Measure-valued Markov processes, Preliminary manuscript. École d’Été de Probabilités de Saint-Flour.

    Google Scholar 

  6. Dawson, D. A., Fleischmann, K. and Gorostiza, L. G. (1989). Stable hydrodynamic limit fluctuations of a critical branching particle system in a random medium, Ann. Prob. 17, 1083–1117.

    Article  Google Scholar 

  7. Dawson, D. A. and Gorostiza, L. G. (1990). Generalized solutions of a class of nuclear space-valued stochastic evolution equations, Appl Math. Optim. 22, 241–263.

    Article  Google Scholar 

  8. Dawson, D. A. and Perkins, E. A. (1991). Historical processes, Mem. Amer. Math. Soc. 454.

    Google Scholar 

  9. Dynkin, E. B. (1991). Branching particle systems and superprocesses, Ann. Prob. 19, 1157–1194.

    Article  Google Scholar 

  10. Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes, Characterization and Convergence. Wiley, New York.

    Google Scholar 

  11. Fernández, B. and Gorostiza, L. G. (1991). A criterion of convergence of generalized processes and an application to a supercritical branching particle system, Canadian J. Math. 43, 1–13.

    Article  Google Scholar 

  12. Fernández, B. and Gorostiza, L. G. (1991). Convergence of generalized semimartingales to a continuous process, in Stochastic Partial Differential Equations and Applications III, (Da Prato, G. and Tubaro, L., Eds.), Pitman Research Notes in Mathematics (to appear).

    Google Scholar 

  13. Fernández, B. and Gorostiza, L. G. (1991). Stability of a class of transformations of distribution-valued processes and stochastic evolution equations, J. Theor. Prob. (to appear).

    Google Scholar 

  14. Gorostiza, L. G. and Löpez-Mimbela, J. A. (1990). The multitype measure branching process, Adv. Appl. Prob. 22, 49–67.

    Article  Google Scholar 

  15. Henry, D. (1981). Geometric Theory of Semilinear Parabolic Equations. Lect. Notes in Math. 840, Springer, Berlin.

    Google Scholar 

  16. Iscoe, I. (1988). A weighted occupation time for a class of measure valued branching processes, Prob. Th. Rel. Fields 71, 85–116.

    Article  Google Scholar 

  17. Joffe, A. and Metivier, M. (1986). Weak convergence of sequences of semimartingales with applications to multitype branching processes. Adv. Appl. Prob. 18, 20–65.

    Article  Google Scholar 

  18. Jakubowski, A. (1986). On the Skorokhod topology, Ann. Inst. H. Poincaré, Sect. B 22, no. 3, 263–285.

    Google Scholar 

  19. Méléard, S. and Roelly, S. (1991). Discontinuous measure-valued branching processes and generalized stochastic equations, Math. Nachr. 154, 141–156.

    Article  Google Scholar 

  20. Roelly-Coppoletta, S. (1986). A criterion of convergence of measure-valued processes: application to measure branching processes, Stochastics 17, 43–65.

    Article  Google Scholar 

  21. Roelly, S. and Rouault, A. (1990). Construction et propiétés de martingales des branchements spatiaux interactifs, Internat. Stat. Rev. 58, 173–189.

    Article  Google Scholar 

  22. Vaillancourt, J. (1990). Interacting Fleming-Viot processes, Stoch. Proc. Appl. 36, 45–57.

    Article  Google Scholar 

  23. Vaillancourt, J. (1990). On the scaling theorem for interacting Fleming-Viot processes, Stoch. Proc. Appl. 36, 263–267.

    Article  Google Scholar 

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© 1993 Springer Basel AG

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Gorostiza, L.G., López-Mimbela, J.A. (1993). A Convergence Criterion for Measure-Valued Processes, and Application to Continuous Superprocesses. In: Nualart, D., Solé, M.S. (eds) Barcelona Seminar on Stochastic Analysis. Progress in Probability, vol 32. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8555-3_4

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  • DOI: https://doi.org/10.1007/978-3-0348-8555-3_4

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9677-1

  • Online ISBN: 978-3-0348-8555-3

  • eBook Packages: Springer Book Archive

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