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On the central limit Newman theorem
Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 382-390 Cet article a éte moissonné depuis la source Math-Net.Ru

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We construct an example of a strictly stationary associated random sequence which does not satisfy the central limit theorem and whose the partial sums' variance grows in a defined regular way. This result generalizes the well-known example of N. Herrndorf and shows the optimality of conditions in the classical Newman's theorem.
Keywords: associated random variables, stationarity, central limit theorem, slowly varying functions. 
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A. P. Shashkin. On the central limit Newman theorem. Teoriâ veroâtnostej i ee primeneniâ, Tome 50 (2005) no. 2, pp. 382-390. http://geodesic.mathdoc.fr/item/TVP_2005_50_2_a12/

[1] Newman C. M., “Normal fluctuations and FKG inequalities”, Comm. Math. Phys., 74:2 (1980), 119–128 | DOI | MR | Zbl

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

[3] Bulinskii A. V., “Funktsionalnyi zakon povtornogo logarifma dlya assotsiirovannykh sluchainykh polei”, Fundam. prikl. matem., 1:3 (1995), 623–639 | MR

[4] Bulinski A. V., “On the convergence rates in the CLT for positively and negatively dependent random fields”, Probability Theory and Mathematical Statistics, eds. I. A. Ibragimov and A. Yu. Zaitsev, Gordon and Breach, Amsterdam, 1996, 3–14 | MR | Zbl

[5] Dabrowski A. R., Jakubowski A., “Stable limits for associated random variables”, Ann. Probab., 22:1 (1994), 1–16 | DOI | MR | Zbl

[6] Herrndorf N., “An example on the central limit theorem for associated sequences”, Ann. Probab., 12:3 (1984), 912–917 | DOI | MR | Zbl

[7] Ibragimov I. A., Linnik Yu. V., Nezavisimye i statsionarno svyazannye velichiny, Nauka, M., 1965, 524 pp.

[8] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977, 351 pp. | MR

[9] Yakymiv A. L., “Mnogomernye tauberovy teoremy i ikh prilozhenie k vetvyaschimsya protsessam Bellmana–Kharrisa”, Matem. sb., 115:3 (1981), 463–477 | MR | Zbl