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Weak convergence of sequences of semimartingales with applications to multitype branching processes

Published online by Cambridge University Press:  01 July 2016

A. Joffe  and
M. Metivier
A. Joffe*
Affiliation:
Université de Montréal
M. Metivier*
Affiliation:
École Polytechnique, Palaiseau
*
∗Postal address: Département de Mathématiques et Statistique, Université de Montréal, P.O. Box 6128, Succ. A, Montréal, Québec H3C 3J7, Canada.
∗Postal address: Département de Mathématiques et Statistique, Université de Montréal, P.O. Box 6128, Succ. A, Montréal, Québec H3C 3J7, Canada.
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Abstract

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

Keywords

WEAK CONVERGENCETIGHTNESSSEMIMARTINGALESD-SEMIMARTINGALESMULTITYPE BRANCHING PROCESS
Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 1 , March 1986 , pp. 20 - 65
Copyright
Copyright © Applied Probability Trust 1986 

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Footnotes

Research supported in part by the Natural Science and Engineering Research Council, Canada, the FCAC Programme of the Ministère de l'Education du Québec, and the Air Force Office of Scientific Research Grant No. F49620–82-C-0009, USA.

References

Aldous, D. (1978) Stopping times and tightness. Ann. Prob. 6, 335–340.CrossRefGoogle Scholar
Aldous, D. (1981) Weak convergence and the general theory of process. Preprint, University of California, Berkeley.Google Scholar
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Billingsley, P. (1969) Convergence of Probability Measures. Wiley, New York.Google Scholar
Billingsley, P. (1974) Conditional distributions and tightness. Ann. Prob. 2, 480–485.CrossRefGoogle Scholar
Buckholtz, P. G. and Wasan, M. T. (1982) Diffusion approximation of the two types Galton-Watson process with mean matrix close to the identity matrix. J. Multivariate Anal. 12, 493–507.CrossRefGoogle Scholar
Dellacherie, C. and Meyer, P. A. (1979) Probabilités et potentiel II. Hermann, Paris.Google Scholar
Feller, W. (1951) Diffusion in genetics. Proc. 2nd Berkeley Symp. Math. Statist. Prob. , 227–246.Google Scholar
Grigelionis, B. (1973) On relative compactness of sets of probability measures in D[0,8[(R). Lit. Math. Sb. 13, 83–96.Google Scholar
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Jacod, J. (1979) Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Jacod, J., Menin, J. and Metivier, M. (1983) On tightness and stopping times. Stoch. Proc. Appl. 14, 109–141.CrossRefGoogle Scholar
Jagers, P. (1971) Diffusion approximation of branching processes. Ann. Math. Statist. 42, 2074–2078.CrossRefGoogle Scholar
Jirina, M. (1969) On Feller's branching diffusion processes. Casopis Pest. Mat. 94, 84–90.CrossRefGoogle Scholar
Kurtz, T. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618–642.CrossRefGoogle Scholar
Kurtz, T. (1978) Diffusion approximations of branching processes. In Advances in Probability 5, ed. Joffe, A. and Ney, P., Dekker, New York, 262–292.Google Scholar
Kurtz, T. (1981) Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36, SIAM, Philadelphia.CrossRefGoogle Scholar
Lamperti, J. (1962) Limiting distributions for branching processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2(2), 225–242.Google Scholar
Lamperti, J. (1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271–288.CrossRefGoogle Scholar
Lenglart, E. (1977) Relations de domination entre deux processus. Ann. Inst. H. Poincaré B 13, 171–179.Google Scholar
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the space D[0, 8[. J. Appl. Prob. 10, 109–121.CrossRefGoogle Scholar
Lipow, C. (1977) Limiting diffusions for population-size dependent branching processes. J. Appl. Prob. 14, 14–24.CrossRefGoogle Scholar
Metivier, ?. (1979) On weak convergence of processes. Internal report, Dept of Mathematics, University of Minnesota, Minneapolis.Google Scholar
Métivier, M. (1982) Semimartingales. De Gruyter, Berlin.CrossRefGoogle Scholar
Metivier, ?. (1984) Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev. Rapport interne de l'Ecole Polytechnique.Google Scholar
Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.CrossRefGoogle Scholar
Platen, E. and Rebolledo, R. (1981) Weak convergence of semi-martingales and discretisation methods. Akad. Wiss. DDR. Preprint P. Maths 15181.Google Scholar
Rebolledo, R. (1979) La méthode des martingales appliquée à la convergence en loi des processus. Mem. Soc. Math. France Suppl. , 1–125.Google Scholar
Skorokhod, S. V. (1956) Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261–290.CrossRefGoogle Scholar
Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.Google Scholar
Whitt, W. (1980) Some useful functions for functional limit theorems. Maths. Operat. Res. 5, 67–85.CrossRefGoogle Scholar