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

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1. [1] N.H Bingham, C.M Goldie, J.L Teugels, Regular Variation, Cambridge, Cambridge University Press, 1987. Zbl0617.26001MR898871
2. [2] D.A Darling, M Kac, On occupation times for Markov processes, Trans. Amer. Math. Soc.84 (1957) 444-458. Zbl0078.32005MR84222
3. [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. MR822069
4. [4] C Dellacherie, P.A Meyer, Probabilités et Potentiel, Chapitres XII à XVI, Hermann, Paris, 1987. Zbl0624.60084MR488194
5. [5] N El Karoui, S Peng, M.C Quenez, Backward stochastic differential equations in finance, J. Math. Finance7 (1) (1997) 1-71. Zbl0884.90035MR1434407
6. [6] A.M Etheridge, Asymptotic behaviour of measure-valued critical branching processes, Proc. Amer. Math. Soc.118 (4) (1993) 1251-1261. Zbl0776.60106MR1100650
7. [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. Zbl0811.60068MR1278288
8. [8] J Hájek, A characterization of limiting distributions of regular estimates, Z. Wahrscheinlichkeitsth. Verw. Geb.14 (1970) 323-330. Zbl0193.18001MR283911
9. [9] R Höpfner, On limits of some martingales arising in recurrent Markov chains, 1988, Unpublished note. 
10. [10] R Höpfner, On statistics of Markov step processes: representation of log-likelihood ratio processes in filtered local models, Probab. Theory Related Fields94 (1993) 375-398. Zbl0766.62051MR1198653
11. [11] R Höpfner, M Hoffmann, E Löcherbach, Non-parametric estimation of the death rate in branching diffusions, Preprint No. 577, University Paris VI, 2000. Zbl1035.62085MR1988418
12. [12] R Höpfner, E Löcherbach, On invariant measure for branching diffusions, 1999, Unpublished note, see http://www.mathematik.uni-mainz.de/~hoepfner. Zbl0997.60092
13. [13] R Höpfner, E Löcherbach, Limit theorems for null recurrent Markov processes, Preprint 22, University of Mainz, 2000, see http://www.mathematik.uni-mainz.de/preprints/2200.html. Zbl1018.60074MR1949295
14. [14] I.A Ibragimov, R.Z Khas'minskii, Statistical Estimation. Asymptotic Theory, Springer, Berlin, 1981. Zbl0467.62026MR620321
15. [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 A17 (1970) 187-214. Zbl0205.43702MR277922
16. [16] J Jacod, A.N Shiryaev, Limit Theorems for Stochastic Processes, Springer, Berlin, 1987. Zbl0635.60021MR959133
17. [17] P Jeganathan, On the asymptotic theory of estimation when the limit of the log-likelihood ratio is mixed normal, Sankhya44 (1982) 173-212. Zbl0584.62042MR688800
18. [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. Zbl0121.35306MR137180
19. [19] L LeCam, G Yang, Asymptotics in Statistics, Springer, New York, 1990. Zbl0719.62003MR1066869
20. [20] R.S Liptser, A.N Shiryaev, Statistics of Random Processes, Vols. 1, 2, Springer, New York, 1978. Zbl1008.62072
21. [21] E Löcherbach, Likelihood ratio processes for Markovian particle systems with killing and jumps, Preprint No. 551, University Paris VI, 1999. MR1917290
22. [22] H Luschgy, Local asymptotic mixed normality for semimartingale experiments, Probab. Theory Related Fields92 (1992) 151-176. Zbl0768.62067MR1161184
23. [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. Zbl0864.60077
24. [24] A.G Pakes, On Markov branching processes with immigration, Sankhya Ser. A37 (1975) 129-138. Zbl0336.60075MR433622
25. [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. Zbl0766.60079
26. [26] S Resnick, P Greenwood, A bivariate stable charakterization and domains of attraction, J. Multivar. Anal.9 (1979) 206-221. Zbl0409.62038MR538402
27. [27] H Strasser, Mathematical Theory of Statistics, de Gruyter, Berlin, 1985. Zbl0594.62017MR812467
28. [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. Zbl0732.60114MR1108185
29. [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. Zbl0627.60069
30. [30] A Wakolbinger, Limits of spatial branching processes, Bernoulli1 (1995) 171-189. Zbl0868.60069MR1354460
31. [31] A.M Zubkov, Life-periods of a branching process with immigration, Theor. Probab. Appl.17 (1972) 174-183. Zbl0267.60084MR300351

1. Reinhard Höpfner, Eva Löcherbach, Remarks on ergodicity and invariant occupation measure in branching diffusions with immigration