View More View Less
  • 1 Université Bordeaux 1 IMB, UMR 5251 351 cours de la Libération 33405 Talence Cedex France
  • 2 Université Paul Sabatier IMT, UMR 5219 118 Route de Narbonne 31062 Toulouse Cedex France
Restricted access

Purchase article

USD  $25.00

1 year subscription (Individual Only)

USD  $800.00

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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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 ) -.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar
  • 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.

    • Search Google Scholar