We prove the almost sure central limit theorem for martingales via an original approach which uses the Carleman moment theorem together with the convergence of moments of martingales. Several statistical applications to autoregressive and branching processes are also provided.
Berkes, I. and Csáki, E. , A universal result in almost sure central limit theory, Stochastic Process. Appl. 94 (2001), 105–134. MR 2002j :60033
Csáki E. , 'A universal result in almost sure central limit theory ' (2001 ) 94 Stochastic Process. Appl. : 105 -134.
Bercu, B. , On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications, Stochastic Process. Appl. 111 (2004), 157–173. MR 2005f :60075
Bercu B. , 'On the convergence of moments in the almost sure central limit theorem for martingales with statistical applications ' (2004 ) 111 Stochastic Process. Appl. : 157 -173.
Billingsley, P. , Probability and measure , third edition John Wiley & Sons, New York, 1995. MR 95k :60001
Billingsley P. , '', in Probability and measure , (1995 ) -.
Brosamler, G. A. , An almost everywhere central limit theorem, Math. Proc. Cambridge Philos. Soc. 104 (1988), 561–574. MR 89i :60045
Brosamler G. A. , 'An almost everywhere central limit theorem ' (1988 ) 104 Math. Proc. Cambridge Philos. Soc. : 561 -574.
Chaabane, F. , Version forte du théorème de la limite centrale fonctionnel pour les martingales, C.R. Acad. Sci. Paris 323 (1996), 195–198. MR 97d :60057
Chaabane F. , 'Version forte du théorème de la limite centrale fonctionnel pour les martingales ' (1996 ) 323 C.R. Acad. Sci. Paris : 195 -198.
Chaabane, F. and Maaouia, F. , Théorèmes limites avec poids pour les martingales vectorielles, Esaim Prob. Stat. 4 (2000), 137–189. MR 2002f :60035
Maaouia F. , 'Théorèmes limites avec poids pour les martingales vectorielles ' (2000 ) 4 Esaim Prob. Stat. : 137 -189.
Chaabane, F. , Invariance principle with logarithm averaging for martingales, Studia Math. Sci. Hungar. 37 (2001), 21–52. MR 2002b :60030
Chaabane F. , 'Invariance principle with logarithm averaging for martingales ' (2001 ) 37 Studia Math. Sci. Hungar. : 21 -52.
Duflo, M. , Random Iterative Models , Springer Verlag, Berlin, 1997. MR 98m :62239
Duflo M. , '', in Random Iterative Models , (1997 ) -.
Feller, W. , An introduction to probability theory and its applications , vol. II, John Wiley, New York, 1966. MR 35 #1048
Feller W. , '', in An introduction to probability theory and its applications, vol. II , (1966 ) -.
Guttorp, P. , Statistical Inference for Branching Processes , John Wiley, New York, 1991. MR 94m :62214
Guttorp P. , '', in Statistical Inference for Branching Processes , (1991 ) -.
Ibragimov, I. A. and Lifshits, M. A. , On the convergence of generalized moments in almost sure central limit theorem, Statist. Probab. Lett. 40 (1998), 343–351. MR 99m :60032
Lifshits M. A. , 'On the convergence of generalized moments in almost sure central limit theorem ' (1998 ) 40 Statist. Probab. Lett. : 343 -351.
Ibragimov, I. A. and Lifshits, M. A. , On almost sure limit theorems, Theory Probab. Appl. 44 (2000), 254–272. MR 2001g :60066
Lifshits M. A. , 'On almost sure limit theorems ' (2000 ) 44 Theory Probab. Appl. : 254 -272.
Lacey, M. T. and Phillip, W. , A note on the almost sure central limit theorem, Statist. Probab. Lett. 9 (1990), 201–205. MR 91e :60100
Phillip W. , 'A note on the almost sure central limit theorem ' (1990 ) 9 Statist. Probab. Lett. : 201 -205.
Lai, T. L. and Wei, C. Z. , Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems, Ann. Statist. 10 (1982), 154–166. MR 84c :62091
Wei C. Z. , 'Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems ' (1982 ) 10 Ann. Statist. : 154 -166.
Lai, T. L. and Wei, C. Z. , Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters, J. Multivariate Analysis 13 (1983), 1–23.
Wei C. Z. , 'Asymptotic properties of general autoregressive models and strong consistency of least-squares estimates of their parameters ' (1983 ) 13 J. Multivariate Analysis : 1 -23.
Lifshits, M. , Almost sure limit theorem for martingales, in: Limit Theorems in Probability and Statistics II (I. Berkes, E. Csáki, M. Csörgő, eds.) J. Bolyai Mathematical Society, Budapest (2002), 367–390. MR 2004c :60090
Lifshits M. , '', in Limit Theorems in Probability and Statistics II , (2002 ) -.
Lifshits, M. , Lecture notes on almost sure limit theorems, Publications IRMA 54 (2001), Lille, 1–25.
Lifshits M. , 'Lecture notes on almost sure limit theorems ' (2001 ) 54 Publications IRMA : 1 -25.
Maaouia, F. and Touati, A. , Identification of multitype branching processes, Ann. Statist. 33 (2005), 2655–2694. MR 2253098
Touati A. , 'Identification of multitype branching processes ' (2005 ) 33 Ann. Statist. : 2655 -2694.
Schatte, P. , On strong versions of the almost sure central limit theorem, Math Nachr. 137 (1988), 249–256. MR 89i :60070
Schatte P. , 'On strong versions of the almost sure central limit theorem ' (1988 ) 137 Math Nachr. : 249 -256.
Schatte, P. , On the central limit theorem with almost sure convergence, Probab. Math. Statist. 11 (1991), 237–246. MR 92k :60050
Schatte P. , 'On the central limit theorem with almost sure convergence ' (1991 ) 11 Probab. Math. Statist. : 237 -246.
Wei, C. Z. , Adaptive prediction by least squares predictors in stochastic regression models with applications to time series, Ann. Statist. 15 (1987), 1667–1682. MR 82e :62123
Wei C. Z. , 'Adaptive prediction by least squares predictors in stochastic regression models with applications to time series ' (1987 ) 15 Ann. Statist. : 1667 -1682.