# zbMATH — the first resource for mathematics

Central limit theorem by moments. (English) Zbl 1133.60311
Summary: In a previous central limit theorem for moments, it has been proved that the moments converge to those of the normal distribution if the moments of the sums are asymptotically independent [cf. R. Blacher, C. R. Acad. Sci., Paris, Sér. I 311, 465–468 (1990; Zbl 0706.60025)]. In this paper we generalize this result by adding a negligible sequence to these sums. So, we can prove that the moments of some functionals of strong mixing sequences converge.

##### MSC:
 60F05 Central limit and other weak theorems
Full Text:
##### References:
 [1] Bernstein, S., Quelques remarques sur le théoréme limite liapounoff, Dokl. akad. nauk SSSR, 24, 3-8, (1939) · JFM 65.0556.01 [2] Birkel, T., Moment bounds for associated sequences, Ann. probab., 16, 3, 1184-1193, (1988) · Zbl 0647.60039 [3] Blacher, R., Theoreme de la limite centrale par LES moments, C. R. acad. sci. Paris, 311, I, 465-468, (1990) · Zbl 0706.60025 [4] Blacher, R., Higher order correlation coefficients, Statistics, 25, 1-15, (1993) · Zbl 0811.62059 [5] Blacher, R., Central limit theorem by polynomial dependence coefficients, J. comput. appl. math., 57, 45-56, (1995) · Zbl 0830.60016 [6] Bradley, R.C., On a very weak bernouilli condition, Stochastics, 13, 61-81, (1984) · Zbl 0538.60036 [7] Brown, B.M., Characteristics functions, moments and the central limit theorem, Ann. math. statist., 41, 658-664, (1970) · Zbl 0196.21204 [8] Cogburn, R., Asymptotic properties of stationary sequences, Univ. California publ. statist., 3, 99-146, (1960) [9] Cox, D.; Kim, T.Y., Moment bounds for mixing random variables useful in nonparametric function estimation, Stochastics process. appl., 56, 151-158, (1995) · Zbl 0817.62027 [10] Dehling, H.; Denker, M.; Phillipps, W., Versik processes and very weak bernouilli processes with summable rates are independent, Proc. amer. math soc., 91, 618-624, (1984) · Zbl 0601.60001 [11] Esseen, C.G.; Janson, S., On moments conditions for normed sums of independent random variables and martingales differences, Stochastics process. appl., 19, 173-182, (1985) · Zbl 0554.60050 [12] Herrndorf, N., A functional central limit theorem for $$\rho$$-mixing sequences, J. multivariate anal., 15, 141-146, (1984) · Zbl 0547.60030 [13] Ibragimov, I.A.; Linnik, Yu.V., Independent and stationary sequences of random variables, (1971), Wolters-Noordhoff Groningen · Zbl 0219.60027 [14] Ibragimov, I.; Lifshits, M., On the convergence of generalized moments in almost sure central limit theorem, Statist. probab. lett., 40, 4, 343-351, (1998) · Zbl 0933.60017 [15] Krugov, V.M., The convergence of moments of random sums, Theory probab. appl., 33, 2, 339-342, (1988) · Zbl 0666.60035 [16] Lancaster, H.O., Orthogonal models for contingency tables. developments in statistics, (1960), Academic Press New York · Zbl 0093.16001 [17] Mairoboda, R.E., The central limit theorem for empirical moment generating functions, Theory probab. appl., 34, 2, 332-335, (1989) · Zbl 0697.60028 [18] Pinsker, M.S., Information and information stability of random variables and processes, (1964), Holden-Day San Francisco · Zbl 0125.09202 [19] Rosenblatt, M., Uniform ergodicity and strong mixing, Z. wahrsch. verw. gebiete., 24, 79-84, (1972) · Zbl 0231.60050 [20] Rozovsky, L.V., An estimate of the remainder in the central limit theorem for a sum of independent random variables with infinite moments of a higher order, Theory probab. appl., 47, 1, 174-183, (2002) · Zbl 1033.60030 [21] Soulier, P., Moment bounds and the central limit theorem for functions of Gaussian vectors, Statist. probab. lett., 54, 2, 193-203, (2001) · Zbl 0993.60019 [22] Withers, C.S., Central limit theorems for dependent random variables, I. Z. wahrsch. verw. gebiete, 54, 509-534, (1981) · Zbl 0451.60027 [23] Yokoyama, R., Moment bounds for stationary mixing sequences, Z. wahrsch. verw. gebiete, 52, 45-57, (1980) · Zbl 0407.60002 [24] Yokoyama, R., The convergence of moments in the central limit theorem for stationary $$\phi$$-mixing processes, Anal. math., 9, 79-84, (1983) · Zbl 0521.60024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.