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

  • A. Joffe (a1) and M. Metivier (a2)

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.

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Postal address: Département de Mathématiques et Statistique, Université de Montréal, P.O. Box 6128, Succ. A, Montréal, Québec H3C 3J7, Canada.
∗∗Postal address: Centre de Mathématiques Appliquées, École Polytechnique, Cedex 91128, Palaiseau, France.

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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.

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References

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Aldous, D. (1978) Stopping times and tightness. Ann. Prob. 6, 335340.
Aldous, D. (1981) Weak convergence and the general theory of process. Preprint, University of California, Berkeley.
Athreya, K. B. and Ney, P. (1972) Branching Processes. Springer-Verlag, Berlin.
Billingsley, P. (1969) Convergence of Probability Measures. Wiley, New York.
Billingsley, P. (1974) Conditional distributions and tightness. Ann. Prob. 2, 480485.
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, 493507.
Dellacherie, C. and Meyer, P. A. (1979) Probabilités et potentiel II. Hermann, Paris.
Feller, W. (1951) Diffusion in genetics. Proc. 2nd Berkeley Symp. Math. Statist. Prob. , 227246.
Grigelionis, B. (1973) On relative compactness of sets of probability measures in D[0,8[(R). Lit. Math. Sb. 13, 8396.
Harris, T. E. (1963) The Theory of Branching Processes. Springer-Verlag, Berlin.
Jacod, J. (1979) Calcul stochastique et problèmes de martingales. Lecture Notes in Mathematics 714, Springer-Verlag, Berlin.
Jacod, J., Menin, J. and Metivier, M. (1983) On tightness and stopping times. Stoch. Proc. Appl. 14, 109141.
Jagers, P. (1971) Diffusion approximation of branching processes. Ann. Math. Statist. 42, 20742078.
Jirina, M. (1969) On Feller's branching diffusion processes. Casopis Pest. Mat. 94, 8490.
Kurtz, T. (1975) Semigroups of conditioned shifts and approximation of Markov processes. Ann. Prob. 3, 618642.
Kurtz, T. (1978) Diffusion approximations of branching processes. In Advances in Probability 5, ed. Joffe, A. and Ney, P., Dekker, New York, 262292.
Kurtz, T. (1981) Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36, SIAM, Philadelphia.
Lamperti, J. (1962) Limiting distributions for branching processes. Proc. 5th Berkeley Symp. Math. Statist. Prob. 2(2), 225242.
Lamperti, J. (1967) The limit of a sequence of branching processes. Z. Wahrscheinlichkeitsth. 7, 271288.
Lenglart, E. (1977) Relations de domination entre deux processus. Ann. Inst. H. Poincaré B 13, 171179.
Lindvall, T. (1973) Weak convergence of probability measures and random functions in the space D[0, 8[. J. Appl. Prob. 10, 109121.
Lipow, C. (1977) Limiting diffusions for population-size dependent branching processes. J. Appl. Prob. 14, 1424.
Metivier, ?. (1979) On weak convergence of processes. Internal report, Dept of Mathematics, University of Minnesota, Minneapolis.
Métivier, M. (1982) Semimartingales. De Gruyter, Berlin.
Metivier, ?. (1984) Convergence faible et principe d'invariance pour des martingales à valeurs dans des espaces de Sobolev. Rapport interne de l'Ecole Polytechnique.
Parthasarathy, K. R. (1967) Probability Measures on Metric Spaces. Academic Press, New York.
Platen, E. and Rebolledo, R. (1981) Weak convergence of semi-martingales and discretisation methods. Akad. Wiss. DDR. Preprint P. Maths 15181.
Rebolledo, R. (1979) La méthode des martingales appliquée à la convergence en loi des processus. Mem. Soc. Math. France Suppl. , 1125.
Skorokhod, S. V. (1956) Limit theorems for stochastic processes. Theory Prob. Appl. 1, 261290.
Stroock, D. W. and Varadhan, S. R. S. (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin.
Whitt, W. (1980) Some useful functions for functional limit theorems. Maths. Operat. Res. 5, 6785.

Keywords

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

  • A. Joffe (a1) and M. Metivier (a2)

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