Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-26T16:39:41.223Z Has data issue: false hasContentIssue false
  • English
  • Français

A Criterion of Convergence of Generalized Processes and an Application to a Supercritical Branching Particle System

Published online by Cambridge University Press:  20 November 2018

Begoña Fernández  and
Luis G. Gorostiza
Begoña Fernández
Affiliation:
Facultad de Ciencias, UNAM, México 04510, D.F. México
Luis G. Gorostiza
Affiliation:
Centro de Investigación y de Estudios Avanzados, México 07300, D.F. México
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of convergence in distribution of a large class of generalized semimartingales to a continuous process is considerably simplified by a recent theorem of Aldous [1], in conjunction with a result of Cremers and Kadelka [3] on convergence of integral functional, and the results of Mitoma [15] and Fouque [8] for generalized processes. We will give a convenient convergence criterion in this setting. The proof amounts to a direct combination of the results of the abovementioned authors, requiring only a minor extension (of a special case) of the theorem of Cremers and Kadelka.

Keywords

60F1760G2060J80
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 5 , 01 October 1991 , pp. 985 - 997
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. Aldous, D., Stopping times and tightness, II, Ann. Probab. 17 (1989), 586–595.Google Scholar
2. Bojdecki, T. and Gorostiza, L.G., Langevin equations for S’-valued Gaussian processes and fluctuation limits of infinite particle systems, Probab. Th. Rel. Fields 73 (1986), 227–244.Google Scholar
3. Cremers, H. and Kadelka, D., On weak convergence of integral functions of stochastic processes with applications to processes taking paths in Lp, Stoch. Proc. Appl. 21 (1986), 305–317.Google Scholar
4. Dawson, D.A. and Gorostiza, L.G., Limit theorems for supercritical branching random fields, Math. Nachr. 118 (1984), 19–46.Google Scholar
5. Dawson, D.A. and Gorostiza, L.G., Generalized solutions of a class of nuclear space valued stochastic evolution equations, Appl. Math. Optim. 22 (1990), 241–263.Google Scholar
6. Ethier, S.N. and Kurtz, T.G., Markov Processes. Characterization and Convergence. Wiley, New York, 1986.Google Scholar
7. Fernández, B. and Gorostiza, L.G., Convergence of generalized semimartingales to a continuous process. (Preprint).Google Scholar
8. Fouque, J.P., La convergence en loi pour les processus a valeurs dans un espace nucléaire, Ann. Inst. H. Poincaré, Sect. B,(3) 22 (1984), 225–245.Google Scholar
9. Gorostiza, L.G., Limites gaussiennespour les champs aléatoires ramifies supercritiques, in “Aspects statistiques et aspects physiques des processus gaussiens”, 385–398. Editions du CNRS, Paris, 1981.Google Scholar
10. Gorostiza, L.G., limit theorems for supercritical branching random fields with immigration, (Tech. Rep. 64, Lab. Res. Stat. Probab. Carleton Univ., 1985), Adv. Appl. Math. 9 (1988), 56–86.Google Scholar
11. Gorostiza, L.G. and Kaplan, N., Invariance principle for branching random motions, Bol. Soc. Mat. Mex., 25 (1980), 63–86.Google Scholar
12. Jakubowski, A., On the Skorokhodtopology, Ann. Inst. H. Poincaré, Sect. B(3) 22 (1986), 263–285.Google Scholar
13. López-Mimbela, J.A., Teoremas limites para campos aleatorios ramificados multitipo. Doctoral thesis, Departamento de Matemâticas, CDSfVESTAV, 1989.Google Scholar
14. Mitoma, I., On the norm continuity of S'-valued Gaussian processes, Nagoya Math. J. 82 (1981), 209–220.Google Scholar
15. Mitoma, I., Tightness of probabilities in C([0,1],5’) and D([0,1],), Ann. Probab. 11 (1983), 989–999.Google Scholar