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On the Almost Sure Central Limit Theorem for Vector Martingales: Convergence of Moments and Statistical Applications

Part of: Probability theory on algebraic and topological structures Miscellaneous applications of functional analysis

Published online by Cambridge University Press:  14 July 2016

Bernard Bercu ,
Peggy Cénac  and
Guy Fayolle
Bernard Bercu*
Affiliation:
Université Bordeaux 1 and INRIA Bordeaux Sud-Ouest
Peggy Cénac*
Affiliation:
Université de Bourgogne
Guy Fayolle*
Affiliation:
INRIA Paris-Rocquencourt
*
∗Postal address: Université Bordeaux 1, Institut de Mathématiques de Bordeaux, UMR 5251, France. Email address: [email protected]
∗∗Postal address: Université de Bourgogne, Institut de Mathématiques de Bourgogne, UMR 5584, 9 rue Alain Savary, BP 47870, 21078 Dijon Cedex, France. Email address: [email protected]
∗∗∗Postal address: INRIA CR Paris-Rocquencourt, Domaine de Voluceau, BP 105, 78153 Le Chesnay Cedex, France. Email address: [email protected]
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Abstract

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We investigate the almost sure asymptotic properties of vector martingale transforms. Assuming some appropriate regularity conditions both on the increasing process and on the moments of the martingale, we prove that normalized moments of any even order converge in the almost sure central limit theorem for martingales. A conjecture about almost sure upper bounds under wider hypotheses is formulated. The theoretical results are supported by examples borrowed from statistical applications, including linear autoregressive models and branching processes with immigration, for which new asymptotic properties are established on estimation and prediction errors.

Keywords

Almost-sure central limit theoremvector martingalemomentstochastic regression

MSC classification

Secondary: 60B12: Limit theorems for vector-valued random variables (infinite-dimensional case) 46N30: Applications in probability theory and statistics
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 151 - 169
Copyright
Copyright © Applied Probability Trust 2009 

References

[1] Bercu, B. (2004). {On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications}. Stoch. Process. Appl. 111, 157–173.Google Scholar
[2] Bercu, B. and Fort, J.-C. (2008). {A moment approach for the almost sure central limit theorem for martingales}. Studia Sci. Math. Hung. 45, 139–159.Google Scholar
[3] Brosamler, G. A. (1988). An almost everywhere central limit theorem. Math. Proc. Camb. Phil. Soc. 104, 561–574.Google Scholar
[4] Chaâbane, F. (2001). Invariance principles with logarithmic averaging for martingales. Studia Sci. Math. Hung. 37, 21–52.Google Scholar
[5] Chaâbane, F. and Maâouia, F. (2000). Théorèmes limites avec poids pour les martingales vectorielles. ESAIM Prob. Statist. 4, 137–189.Google Scholar
[6] Chaâbane, F. and Touati, A. (2001). On averaging methods for identification of linear regression models. C. R. Acad. Sci. Paris 333, 133–138.CrossRefGoogle Scholar
[7] Chaâbane, F., Maâouia, F. and Touati, A. (1998). Généralisation du théorème de la limite centrale presque-sûr pour les martingales vectorielles. C. R. Acad. Sci. Paris 326, 229–232.Google Scholar
[8] Duflo, M. (1997). Random Iterative Methods (Appl. Math. 34). Springer, Berlin.Google Scholar
[9] Duflo, M., Senoussi, R. and Touati, A. (1991). {Propriétés asymptotiques presque sûres de l'estimateur des moindres carrés d'un modèle autorégressif vectoriel}. Ann. Inst. H. Poincaré Prob. Statist. 27, 1–25.Google Scholar
[10] Goodwin, G. and Sin, K. (1984). {Adaptive Filtering Prediction and Control.} Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
[11] Lacey, M. T. and Phillip, W. (1990). {A note on the almost sure central limit theorem}. Statist. Prob. Lett. 9, 201–205.Google Scholar
[12] Lai, T. L. and Wei, C. Z. (1982). {Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems}. Ann. Statist. 10, 154–166.Google Scholar
[13] Lai, T. L. and Wei, C. Z. (1983). {Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters}. J. Multivariate Anal. 13, 1–23.Google Scholar
[14] Lifshits, M. A. (2001). Lecture notes on almost sure limit theorems. Publ. IRMA 54, 1–25.Google Scholar
[15] Lifshits, M. A. (2002). Almost sure limit theorem for martingales. In Limit Theorems in Probability and Statistics, Vol. II, János Bolyai Mathematical Society, Budapest, pp. 367–390.Google Scholar
[16] Schatte, P. (1988). On strong versions of the central limit theorem. Math. Nachr. 137, 249–256.Google Scholar
[17] Wei, C. Z. (1985). {Asymptotic properties of least-squares estimates in stochastic regression models}. Ann. Statist. 13, 1498–1508.Google Scholar
[18] Wei, C. Z. (1987). {Adaptative prediction by least squares predictors in stochastic regression models with applications to time series}. Ann. Statist. 15, 1667–1682.Google Scholar
[19] Wei, C. Z. and Winnicki, J. (1990). Estimation of the means in the branching process with immigration. Ann. Statist. 18, 1757–1773.Google Scholar
[20] Winnicki, J. (1991). Estimation of the variances in the branching process with immigration. Prob. Theory Relat. Fields 88, 77–106.Google Scholar