LAN and LAMN for systems of interacting diffusions with branching and immigration

Annales de l'institut Henri Poincare (B) Probability and Statistics

Volumn 38, Issue 1, 2002, Pages 59-90

  Löcherbach, Eva ^a  

^a UNIVERSITÉ PARIS EST CRÉTEIL   (France)

Author keywords

Branching diffusions; Local asymptotic mixed normality; Local asymptotic normality; Particle systems

Indexed keywords

EID: 0011963649     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0246-0203(01)01077-9     Document Type: Article

References (32)

1. Bingham N.H., Goldie C.M., Teugels J.L., Regular Variation, Cambridge, Cambridge University Press, 1987.
2. Darling D.A., Kac M., On occupation times for Markov processes, Trans. Amer. Math. Soc. 84 (1957) 444-458.
3. Davies R.B., Asymptotic inference when the amount of information is random, in: LeCam L., Ohlsen R. (Eds.), Proceedings of the Berkeley Conference in honour of J. Neyman and J. Kiefer, Vol. 2, Monterey, Wadsworth, 1985.
4. Dellacherie C., Meyer P.A., Probabilités et Potentiel, Chapitres XII à XVI, Hermann, Paris, 1987.
5. El Karoui N., Peng S., Quenez M.C., Backward stochastic differential equations in finance, J. Math. Finance 7 (1) (1997) 1-71.
6. Etheridge A.M., Asymptotic behaviour of measure-valued critical branching processes, Proc. Amer. Math. Soc. 118 (4) (1993) 1251-1261.
7. Gorostiza L., Wakolbinger A., Long time behaviour of critical branching particle systems and applications, in: Dawson D.A. (Ed.), Measure Valued Processes, Stochastic Partial Differential Equations and Interacting Systems, CRM Proc. Lect. Notes, Vol. 5, AMS, Providence, 1994, pp. 119-137.
8. Hájek J., A characterization of limiting distributions of regular estimates, Z. Wahrscheinlichkeitsth. Verw. Geb. 14 (1970) 323-330.
9. Höpfner R., On limits of some martingales arising in recurrent Markov chains, 1988. Unpublished note.
10. Höpfner R., On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models, Probab. Theory Related Fields 94 (1993) 375-398.
11. Höpfner R., Hoffmann M., Löcherbach E., Non-parametric estimation of the death rate in branching diffusions, Preprint No. 577, University Paris VI, 2000.
12. Höpfner R., Löcherbach E., On invariant measure for branching diffusions, 1999, Unpublished note, see http://www.mathematik.uni-mainz.de/ ~hoepfner.
13. Höpfner R., Löcherbach E., Limit theorems for null recurrent Markov processes, Preprint 22, University of Mainz, 2000, see http://www.mathematik.uni-mainz.de/preprints/2200.html.
14. Ibragimov I.A., Khas'minskii R.Z., Statistical Estimation. Asymptotic Theory, Springer, Berlin, 1981.
15. Ikeda N., Watanabe S., On uniqueness and non-uniqueness of solutions for a class of non-linear equations and explosion problem for branching processes, J. Fac. Sci. Tokyo, Sect. I A 17 (1970) 187-214.
16. Jacod J., Shiryaev A.N., Limit Theorems for Stochastic Processes, Springer, Berlin, 1987.
17. Jeganathan P., On the asymptotic theory of estimation when the limit of the log-likelihood ratio is mixed normal, Sankhya 44 (1982) 173-212.
18. Karlin S., McGregor J., Occupation time laws for birth and death processes, in: Neyman J. (Ed.), Proc. Fourth Berkeley Symp. Math. Stat. Prob. 2, Berkeley, University of California Press, 1961, pp. 249-273.
19. LeCam L., Yang G., Asymptotics in Statistics, Springer, New York, 1990.
20. Liptser R.S., Shiryaev A.N., Statistics of Random Processes, Vols. 1,2, Springer, New York, 1978.
21. Löcherbach E., Likelihood ratio processes for Markovian particle systems with killing and jumps, Preprint No. 551, University Paris VI, 1999.
22. Luschgy H., Local asymptotic mixed normality for semimartingale experiments, Probab. Theory Related Fields 92 (1992) 151-176.
23. Méléard S., Asymptotic behavior of some interacting particle systems; McKean-Vlasov and Boltzmann models, in: Graham C. et al. (Eds.), Probabilistic Models for Nonlinear Partial Differential Equations, Proc. Montecatini Terme 1995, Lecture Notes in Math., Vol. 1627, Springer, Berlin, 1996, pp. 42-95.
24. Pakes A.G., On Markov branching processes with immigration, Sankhya Ser. A 37 (1975) 129-138.
25. Pardoux E., Peng S., Backward SDE's and quasilinear PDE's, in: Rozovskii B.L., Sowes R.B. (Eds.), Stochastic Partial Differential Equations and their Applications, LNCIS, Vol. 176, Springer, Berlin, 1990.
26. Resnick S., Greenwood P., A bivariate stable charakterization and domains of attraction, J. Multivar. Anal. 9 (1979) 206-221.
27. Strasser H., Mathematical Theory of Statistics, de Gruyter, Berlin, 1985.
28. Sznitman A.S., Topics in Propagation of Chaos, Ecole d'été de Probabilités de Saint Flour XIX 1989, Lecture Notes in Math., Vol. 1464, Springer, New York, 1991.
29. Touati A., Théorèmes limites pour les processus de Markov récurrents, 1988, Unpublished paper. See also C. R. Acad. Sci. Paris Série 1 305 (1987) 841-844.
30. Touati A., Théorèmes limites pour les processus de Markov récurrents, 1988, Unpublished paper. See also C. R. Acad. Sci. Paris Série 1 305 (1987) 841-844.
31. Wakolbinger A., Limits of spatial branching processes, Bernoulli 1 (1995) 171-189.
32. Zubkov A.M., Life-periods of a branching process with immigration, Theor. Probab. Appl. 17 (1972) 174-183.