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Statistical version of the central limit theorem (CLT) with random matrix normalization is established for random fields with values in a space Rk (k ≥ 1). Dependence structure of the field under consideration is described in terms of the covariance inequalities for the class of bounded Lipschitz ”test functions” defined on finite disjoint collections of random vectors constituting the field. The main result provides an estimate of the convergence rate, over a family of convex bounded sets, in the CLT with random normalization.
Czasopismo
Rocznik
Tom
Strony
261--281
Opis fizyczny
Bibliogr. 27 poz.
Twórcy
autor
- Department of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia
autor
- Department of Mathematics and Mechanics, Moscow State University, Moscow 119992, Russia
Bibliografia
- [1] Y. Y. Bakhtin and A. V. Bulinski, Moment inequalities for the sums of dependent multiindexed random variables (in Russian), Fundam. Prikl. Mat. 3 (1997), pp. 1101-1108.
- [2] P. Billingsley, Convergence of Probability Measures, Wiley, New York 1968.
- [3] E. Bolthausen, On the central limit theorem for mixing random fields, Ann. Probab. 10 (1982), pp. 1047-1050.
- [4] A. V. Bulinski, On the convergence rates in the CLT for positively or negatively dependent random fields, in: Probability Theory and Mathematical Statistics, I. A. Ibragimov and A. Yu. Zaitsev (Eds.), Gordon and Breach, 1996, pp. 3-14.
- [5] A. V. Bulinski, Asymptotic Gaussian behavior of quasi-associated vector-valued random fields (in Russian), Obozr. Prikl. and Prom. Mat. 7 (2000), pp. 482-483.
- [6] A. V. Bulinski, Statistical version of the central limit theorem for vector-valued random fields, Math. Notes 76 (2004), pp. 455-464.
- [7] A. Bulinski and A. Khrennikov, Generalization of the critical volume NTCP model in the radiobiology, Université P. et M. Curie Paris-6, CNRS U.M.R. 7599, Probabilités et Modèles Aléatoires, Prépublication, PMA-977 (2005), Preprint, Paris-6, pp. 1-13.
- [8] A. V. Bulinski and E. Shabanovich, Asymptotic behavior for some functionals of positively and negatively dependent random fields (in Russian), Fundam. Prinkl. Mat. 4 (1998), pp. 479-492.
- [9] A. V. Bulinski and A. P. Shashkin, Rates in the central limit theorem for dependent multiindexed random vectors, J. Math. Sci. 122 (2004), pp. 3343-3358.
- [10] A. Bulinski and A. Shashkin, Limit Theorems for Associated Random Fields and Related Systems, 2006, 400 pp. (to appear).
- [11] A. V. Bulinski and A. P. Shashkin, Strong invariance principle for dependent random fields, IMS Lecture Notes Monogr. Ser. Dynamics and Stochastics 48 (2006), pp. 128-143.
- [12] A. Bulinski and C. Suquet, Normal approximation for quasi-associated random fields, Statist. Probab. Lett. 54 (2001), pp. 215-226.
- [13] A. V. Bulinski and M. A. Vronski, Statistical variant of the central limit theorem for associated random fields (in Russian), Fundam. Prikl. Mat. 2 (1996), pp. 999-1018.
- [14] P. Doukhan and G. Lang, Rates in the empirical central limit theorem for stationary weakly dependent random fields, Stat, Inference Stoch. Process. 5 (2002), pp. 199-228.
- [15] P. Doukhan and S. Louhichi, A new weak dependence condition and application to moment inequalities, Stochastic Process. Appl. 84 (1999), pp. 313-342.
- [16] J. Esary, F. Proschan and D. Walkup, Association of random variables with applications, Ann. Math. Statist. 38 (1967), pp. 1466-1474.
- [17] G. M. Fihtengolts, Calculus (in Russian), Fizmatlit, Moscow 2003.
- [18] E. Giné and F. Götze, On standard normal convergence of the multivariate Student t-statistic for symmetric random vectors, Electron. Comm. Probab. 9 (2004), pp. 162-171.
- [19] E. Giné, F. Götze and D. Mason, When is the Student t-statistic asymptotically standard normal?, Ann. Probab. 25 (1997), pp. 1514-1531.
- [20] M. G. Hahn and M. J. Klass, Matrix normalization of sums of random vectors in the domain of attraction of the multivariate normal law, Ann. Probab. 8 (1980), pp. 262-280.
- [21] K. Joag-Dev and F. Proschan, Negative association of random variables, with applications, Ann. Statist. 11 (1983), pp. 286-295.
- [22] R. A. Maller, Quadratic negligibility and the asymptotic normality of operator normed sums, J. Multivariate Anal. 44 (1993), pp. 191-219.
- [23] R. A. Maller, M. J. Klass and H. T. V. Vu, On the Studentization of random vectors, J. Multivariate Anal. 57 (1996), pp. 142-155.
- [24] M. Markus and Kh. Mink, Review of Matrix Theory and Matrix Inequalities (in Russian), Nauka, Moscow 1972.
- [25] C. M. Newman, Asymptotic independence and limit theorems for positively and negatively dependent random variables, in: Inequalities in Statistics and Probability, Y. L. Tong (Ed.), Hayward, 1984, pp. 127-140.
- [26] M. Peligrad and Q.-M. Shao, Self-normalized central limit theorem for sums of weakly dependent random variables, J. Theoret. Probab. 7 (2) (1994), pp. 309-338.
- [27] S. J. Sepański, Probabilistic characterizations of the generalized domain of attraction of the multivariate normal law, J. Theoret. Probab. 7 (1994), pp. 857-866.
Uwagi
To the memory of Professor Kazimierz Urbanik.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-8973e3b8-bd44-4644-8622-e7b8023fe02a