LAN and LAMN for systems of interacting diffusions with branching and immigration
Annales de l'I.H.P. Probabilités et statistiques, Tome 38 (2002) no. 1, pp. 59-90.
@article{AIHPB_2002__38_1_59_0,
author = {L\"ocherbach, Eva},
title = {LAN and LAMN for systems of interacting diffusions with branching and immigration},
journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
pages = {59--90},
publisher = {Elsevier},
volume = {38},
number = {1},
year = {2002},
zbl = {1004.60078},
mrnumber = {1899230},
language = {en},
url = {www.numdam.org/item/AIHPB_2002__38_1_59_0/}
}
Löcherbach, Eva. LAN and LAMN for systems of interacting diffusions with branching and immigration. Annales de l'I.H.P. Probabilités et statistiques, Tome 38 (2002) no. 1, pp. 59-90. http://www.numdam.org/item/AIHPB_2002__38_1_59_0/

[1] N.H Bingham, C.M Goldie, J.L Teugels, Regular Variation, Cambridge, Cambridge University Press, 1987. | MR 898871 | Zbl 0617.26001

[2] D.A Darling, M Kac, On occupation times for Markov processes, Trans. Amer. Math. Soc. 84 (1957) 444-458. | MR 84222 | Zbl 0078.32005

[3] R.B Davies, 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. | MR 822069

[4] C Dellacherie, P.A Meyer, Probabilités et Potentiel, Chapitres XII à XVI, Hermann, Paris, 1987. | MR 488194 | Zbl 0624.60084

[5] N El Karoui, S Peng, M.C Quenez, Backward stochastic differential equations in finance, J. Math. Finance 7 (1) (1997) 1-71. | MR 1434407 | Zbl 0884.90035

[6] A.M Etheridge, Asymptotic behaviour of measure-valued critical branching processes, Proc. Amer. Math. Soc. 118 (4) (1993) 1251-1261. | MR 1100650 | Zbl 0776.60106

[7] L Gorostiza, A Wakolbinger, 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, 5, AMS, Providence, 1994, pp. 119-137. | MR 1278288 | Zbl 0811.60068

[8] J Hájek, A characterization of limiting distributions of regular estimates, Z. Wahrscheinlichkeitsth. Verw. Geb. 14 (1970) 323-330. | MR 283911 | Zbl 0193.18001

[9] R Höpfner, On limits of some martingales arising in recurrent Markov chains, 1988, Unpublished note.

[10] R Höpfner, On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models, Probab. Theory Related Fields 94 (1993) 375-398. | MR 1198653 | Zbl 0766.62051

[11] R Höpfner, M Hoffmann, E Löcherbach, Non-parametric estimation of the death rate in branching diffusions, Preprint No. 577, University Paris VI, 2000. | MR 1988418

[12] R Höpfner, E Löcherbach, On invariant measure for branching diffusions, 1999, Unpublished note, see .

[13] R Höpfner, E Löcherbach, Limit theorems for null recurrent Markov processes, Preprint 22, University of Mainz, 2000, see . | MR 1949295

[14] I.A Ibragimov, R.Z Khas'Minskii, Statistical Estimation. Asymptotic Theory, Springer, Berlin, 1981. | MR 620321 | Zbl 0467.62026

[15] N Ikeda, S Watanabe, 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. | MR 277922 | Zbl 0205.43702

[16] J Jacod, A.N Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. | MR 959133 | Zbl 0635.60021

[17] P Jeganathan, On the asymptotic theory of estimation when the limit of the log-likelihood ratio is mixed normal, Sankhya 44 (1982) 173-212. | MR 688800 | Zbl 0584.62042

[18] S Karlin, J Mcgregor, 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. | MR 137180 | Zbl 0121.35306

[19] L Lecam, G Yang, Asymptotics in Statistics, Springer, New York, 1990. | MR 1066869 | Zbl 0719.62003

[20] R.S Liptser, A.N Shiryaev, Statistics of Random Processes, Vols. 1,2, Springer, New York, 1978. | Zbl 1008.62072

[21] E Löcherbach, Likelihood ratio processes for Markovian particle systems with killing and jumps, Preprint No. 551, University Paris VI, 1999. | MR 1917290

[22] H Luschgy, Local asymptotic mixed normality for semimartingale experiments, Probab. Theory Related Fields 92 (1992) 151-176. | MR 1161184 | Zbl 0768.62067

[23] S Méléard, Asymptotic behavior of some interacting particle systems; McKean-Vlasov and Boltzmann models, in: Graham C, (Eds.), Probabilistic Models for Nonlinear Partial Differential Equations, Proc. Montecatini Terme 1995, Lecture Notes in Math., 1627, Springer, Berlin, 1996, pp. 42-95. | Zbl 0864.60077

[24] A.G Pakes, On Markov branching processes with immigration, Sankhya Ser. A 37 (1975) 129-138. | MR 433622 | Zbl 0336.60075

[25] E Pardoux, S Peng, Backward SDE's and quasilinear PDE's, in: Rozovskii B.L, Sowes R.B (Eds.), Stochastic Partial Differential Equations and their Applications, LNCIS, 176, Springer, Berlin, 1990. | Zbl 0766.60079

[26] S Resnick, P Greenwood, A bivariate stable charakterization and domains of attraction, J. Multivar. Anal. 9 (1979) 206-221. | MR 538402 | Zbl 0409.62038

[27] H Strasser, Mathematical Theory of Statistics, de Gruyter, Berlin, 1985. | MR 812467 | Zbl 0594.62017

[28] A.S Sznitman, Topics in Propagation of Chaos, Ecole d'été de Probabilités de Saint Flour XIX 1989, Lecture Notes in Math., 1464, Springer, New York, 1991. | MR 1108185 | Zbl 0732.60114

[29] A Touati, Théorèmes limites pour les processus de Markov récurrents, 1988, Unpublished paper. See also C. R. Acad. Sci. Paris Série I 305 (1987) 841-844. | Zbl 0627.60069

[30] A Wakolbinger, Limits of spatial branching processes, Bernoulli 1 (1995) 171-189. | MR 1354460 | Zbl 0868.60069

[31] A.M Zubkov, Life-periods of a branching process with immigration, Theor. Probab. Appl. 17 (1972) 174-183. | MR 300351 | Zbl 0267.60084