Abstract
We consider statistical models for finite systems of branching diffusions with immigrations. We give necessary and sufficient conditions for local absolute continuity of laws for such branching particle systems on a suitable path space and derive an explicit version of the likelihood ratio process. For ergodic parametric submodels, under assumptions which combine smoothness properties of the parametrization at a fixed parameter point ϑ and integrability of certain information processes with respect to the invariant measure of the process under ϑ or with respect to an associated Campbell measure, one deduces local asymptotic normality at ϑ (LAN(ϑ)). Moreover, for null recurrent models, local asymptotic mixed normality (LAMN) at ϑ holds in situations where the right limit theorems for integrable additive functionals of the process are hand. These limit theorems follow from dividing the trajectory into independent pieces between successive returns to the void configuration and from limit theorems to stable processes.
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References
Bingham,N. H.,Goldie, C. M., Teugels,J. L. (1987):Regular variation Cambridge New York Sydney: Cambridge University Press.
Brémaud,P.(1981):Point Processes and queues, martingale dynamics Berlin Heidelberg New York: Springer.
Eisele,Th. (1981):Diffusions avec branchement et interaction. Ann. Sci. Univ. Clermont-Ferrand II 69, Math. 19, 21–34.
Greenwood,P.,Resnick,S.(1979):A bivariate stable characterization and domains of attraction. J. Multivar. Anal. 9, 206–221
Hájek,J. (1970):A characterization of limiting distributions of regular estimates Z. Wahrscheinlichkeitsth. Verw. Geb. 14, 323–330.
Höpfner, R., Hoffmann, M., Löcherbach, E.(2000):Nonparametric estimation of the death rate in branching diffusions Preprint 577, Laboratoire de Probabilités, Université Paris VI.
Höpfner,R. Löcherbach,E,On invariant measure for branching diffusions,Unpublished note,see http://www.mathematik.uni-zmainz.de/~hoepfner/~hoepfner
Ikeda,N.,Nagasawa,M.,Watanabe,S.(1968)Branching Markov processes II. J. Math. Kyoto Univ. 8, 365–410
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.
Jacod,J.(1975):Multivariate point processes: predictable projection,RadonNikodym derivates, representation of martingales. Z. Wahrscheinlichkeitsth. Verw. Geb. 31, 235–253
Jacod,J.,Shiryaev,A. N.(1987):Limit Theorems for Stochastic Processes Berlin Heidelberg New York: Springer.
Jeganathan,P. (1982):On the asymptotic theory of estimation when the limit of the log-likelihood ratio is mixed normal. Sankhya 44, 173–212.
Karlin,S.,McGregor,J.(1961):Occupation time laws for birth and death processes In: Neyman, J. (Ed.), Proc. Fourth Berkeley Symp. Math. Stat. Prob. 2, pp. 249–273. Berkeley, University of California Press
Le Cam,L. (1960):Locally asymptotic normal families of distributions Univ. Calif. Publ. Statistics 3, 37–98.
Löcherbach, E. (1999a): Statistical models and likelihood ratio processes for interacting particle systems with branching and immigration. PhD-Thesis, University of Paderborn.
Löcherbach, E. (1999b): Likelihood ratio processes for Markovian particle systems with killing and jumps.Preprint 551, Laboratoire de Probabilités, Université Paris VI.
Löcherbach, E. (2000): LAN and LAMN for systems of interacting diffusions with branching and immigration.Preprint 590, Laboratoire de Probabilités, Université Paris VI.
Luschgy,H.(1992):Local asymptotic mixed normality for semimartingale experiments.Probab. Theory Relat. Fields. 92, 151–176.
Méléard,S. (1996): Asymptotic behavior of some interacting particle systems; McKean-Vlasov and Boltzmann models. In: Graham, C. (ed.) et al., Probabilistic models for nonlinear partial differential equations, Proc. Montecatini Terme 1995, pp. 42–95. Lect. Notes in Mathematics 1627, Berlin: Springer.
Spohn, H. (1998). Dyson’s model of interacting Brownian motions at arbitrary coupling strength. Markov Process. Relat. Fields 4, 649–661.
Sznitman,A.S.(1991).Topics in propagation of chaos. Ecole d’été de Probabilités de Saint Flour XIX 1989, Lecture Notes in Mathematics 1464, New York: Springer.
Touati,A. (1988):Théorèmes limites pour les processes de Markov recur-rents. Unpublished manuscript
Wakolbinger, A.(1995):Limits of spatial branching processes Bernoulli 1, 171–189.
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Löcherbach, E. (2000). Likelihood ratio processes and asymptotic statistics for systems of interacting diffusions with branching and immigration. In: Gardy, D., Mokkadem, A. (eds) Mathematics and Computer Science. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8405-1_23
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DOI: https://doi.org/10.1007/978-3-0348-8405-1_23
Publisher Name: Birkhäuser, Basel
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