Geodesic
Statistical Version of the Central Limit Theorem for Vector-Valued Random Fields
Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 490-501.

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The classical central limit theorem due to Newman for real-valued strictly stationary associated random fields is generalized to strictly stationary quasi-associated vector-valued random fields comprising, in particular, positively or negatively associated fields with finite second moments. We also establish a version of the CLT with random matrix normalization which allows us to construct approximate confidence intervals for the unknown mean vector. 
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A. V. Bulinski. Statistical Version of the Central Limit Theorem for Vector-Valued Random Fields. Matematičeskie zametki, Tome 76 (2004) no. 4, pp. 490-501. http://geodesic.mathdoc.fr/item/MZM_2004_76_4_a1/

[1] Bulinski A., Suquet C., “Normal approximation for quasi-associated random fields”, Statist. Probability Lett., 54 (2001), 215–226 | DOI | MR | Zbl

[2] Bulinskii A. V., “Asimptoticheskaya normalnost kvazi-assotsiirovannykh vektornykh sluchainykh polei”, Obozrenie prikladn. i promyshl. matem., 7 (2000), 482–483

[3] Doukhan P., Louhichi S., “A new weak dependence condition and application to moment inequalities”, Stoch. Proc. Appl., 84 (1999), 313–342 | DOI | MR | Zbl

[4] Doukhan P., Lang G., “Rates in the empirical central limit theorem for stationary weakly dependent random fields”, Statist. Inf. Stoch. Proc., 5 (2002), 199–228 | DOI | MR | Zbl

[5] Esary J., Proschan F., Walkup D., “Association of random variables with applications”, Ann. Math. Statist., 38 (1967), 1466–1474 | DOI | MR | Zbl

[6] Newman C. M., “Asymptotic independence and limit theorems for positively and negatively dependent random varialbes”, Inequalities in Statistics and Probability, IMS Lecture Notes Monograph Ser., ed. Y. L. Tong, Inst. Math. Statist., Hayward, CA, 1984, 127–140

[7] Joag-Dev K., Proschan F., “Negative association of random variables, with applications”, Ann. Statist., 11 (1983), 286–295 | DOI | MR

[8] Ebrahimi E., “On the dependence structure of certain multidimensional Ito processes and corresponding hitting times”, J. Multivar. Anal., 81 (2002), 128–137 | DOI | MR

[9] Evans S., “Association and random measures”, Probability Theory Related Fields, 86 (1990), 1–19 | DOI | MR | Zbl

[10] Fortuin C., Kasteleyn P., Ginibre J., “Correlation inequalities on some partially ordered sets”, Comm. Math. Phys., 22 (1971), 89–103 | DOI | MR | Zbl

[11] Lee M.-L. T., Rachev S. T., Samorodnitsky G., “Association of stable random variables”, Ann. Probability, 18 (1990), 1759–1764 | DOI | MR | Zbl

[12] Newman C. M., “Normal fluctuations and FKG-inequalities”, Comm. Math. Phys., 74 (1980), 119–128 | DOI | MR | Zbl

[13] Pitt L., “Positively correlated normal variables are associated”, Ann. Probability, 10 (1982), 496–499 | DOI | MR | Zbl

[14] Rachev S. T., Xin H., “Test on association of random variables in the domain of a multivariate stable law”, Probability Math. Statist., 14 (1993), 125–141 | MR | Zbl

[15] Bulinskii A. V., Shabanovich E., “Asimptoticheskoe povedenie nekotorykh funktsionalov ot polozhitelno i otritsatelno zavisimykh sluchainykh polei”, Fundament. i prikl. matem., 4 (1998), 479–492 | MR | Zbl

[16] Birkel T., “On the convergence rate in the central limit theorem for associated processes”, Ann. Probab., 16:4 (1988), 1685–1698 | DOI | MR | Zbl

[17] Roussas G. G., “Asymptotic normality of random fields of positively and negatively associated processes”, J. Multivar. Anal., 50 (1994), 152–173 | DOI | MR | Zbl

[18] Peligrad M., Shao Q.-M., “Estimation of variance of partial sums of an associated sequence of random variables”, Stochastic Proc. Appl., 56 (1995), 307–319 | DOI | MR | Zbl

[19] Bulinski A. V., “On the convergence rates in the central limit theorem for positively and negatively dependent random fields”, Probability Theory and Mathematical Statistics, Proceedings of the Kolmogorov semester in the Euler Math. Inst. (March 1993), eds. I. A. Ibragimov, A. Yu. Zaitsev, Gordon and Breach, London, 1996, 3–14 | MR | Zbl

[20] Burton R., Dabrowski A. R., Dehling H., “An invariance principle for quasi-associated random vectors”, Stochastic Process. Appl., 23 (1986), 301–306 | DOI | MR | Zbl

[21] Bulinski A., Shashkin A., Rates in the CLT for vector-valued random fields, Prépublications de Mathématiques de l'Université Paris 10, No 24, Nanterre, 2003, p. 1–27 | MR

[22] Shashkin A. P., “Kvazi-assotsiirovannost gaussovskikh sistem vektorov”, UMN, 57:6 (2002), 199–200 | MR

[23] Bakhtin Yu. Yu., Bulinskii A. V., “Momentnye neravenstva dlya summ zavisimykh multiindeksirovannykh velichin”, Fundament. i prikl. matem., 3 (1997), 1101–1108 | MR | Zbl

[24] Ryuel D., Statisticheskaya mekhanika, Mir, M., 1972

[25] Bolthausen E., “On the central limit theorem for mixing random fields”, Ann. Probability, 10 (1982), 1047–1050 | DOI | MR | Zbl

[26] Bulinskii A. V., Vronskii M. A., “Statisticheskii variant tsentralnoi predelnoi teoremy dlya assotsiirovannykh sluchainykh polei”, Fundament. i prikl. matem., 2:4 (1996), 999–1018 | MR | Zbl