×

zbMATH — the first resource for mathematics

Normal approximation under local dependence. (English) Zbl 1048.60020
This paper is concerned with normal approximation under local dependence by using Stein’s method. Local dependence roughly means that certain subsets of the random variables are independent of those outside their respective “neighborhoods”. No structure on the index set is assumed. Both uniform and nonuniform error bounds of the Berry-Esseen type are obtained and shown to be more general or sharper than many existing results in the literature.

MSC:
60F05 Central limit and other weak theorems
60G60 Random fields
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Baldi, P. and Rinott, Y. (1989). On normal approximation of distributions in terms of dependency graph. Ann. Probab. 17 1646–1650. JSTOR: · Zbl 0691.60020 · doi:10.1214/aop/1176991178 · links.jstor.org
[2] Baldi, P., Rinott, Y. and Stein, C. (1989). A normal approximation for the number of local maxima of a random function on a graph. In Probability , Statistics , and Mathematics. Papers in Honor of Samuel Karlin (T. W. Anderson, K. B. Athreya and D. L. Iglehart, eds.) 59–81. Academic Press, Boston. · Zbl 0704.62018
[3] Berry, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variables. Trans. Amer. Math. Soc. 49 122–136. · Zbl 0025.34603 · doi:10.2307/1990053
[4] Bikelis, A. (1966). Estimates of the remainder in the central limit theorem. Litovsk. Mat. Sb. 6 323–346.
[5] Chen, L. H. Y. (1978). Two central limit problems for dependent random variables. Z. Wahrsch. Verw. Gebiete 43 223–243. · Zbl 0364.60049 · doi:10.1007/BF00536204
[6] Chen, L. H. Y. (1986). The rate of convergence in a central limit theorem for dependent random variables with arbitrary index set. IMA Preprint Series #243, Univ. Minnesota. · Zbl 0614.60019
[7] Chen, L. H. Y. (1998). Stein’s method: Some perspectives with applications. Probability Towards 2000 . Lecture Notes in Statist. 128 97–122. Springer, Berlin. · Zbl 1044.60505
[8] Chen, L. H. Y. and Shao, Q. M. (2001). A non-uniform Berry–Esseen bound via Stein’s method. Probab. Theory Related Fields 120 236–254. · Zbl 0996.60029 · doi:10.1007/s004400100124
[9] Dasgupta, R. (1992). Nonuniform speed of convergence to normality for some stationary \(m\)-dependent processes. Calcutta Statist. Assoc. Bull. 42 149–162. · Zbl 0770.60022
[10] Dembo, A. and Rinott, Y. (1996). Some examples of normal approximations by Stein’s method. In Random Discrete Structures (D. Aldous and R. Pemantle, eds.) 25–44. Springer, New York. · Zbl 0847.60015
[11] Erickson, R. V. (1974). \(L_1\) bounds for asymptotic normality of \(m\)-dependent sums using Stein’s technique. Ann. Probab. 2 522–529. · Zbl 0301.60020 · doi:10.1214/aop/1176996670
[12] Esseen, C. G. (1945). Fourier analysis of distribution functions: A mathematical study of the Laplace–Gaussian law. Acta Math. 77 1–125. · Zbl 0060.28705 · doi:10.1007/BF02392223
[13] Esseen, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. Verw. Gebiete 9 290–308. · Zbl 0195.19303 · doi:10.1007/BF00531753
[14] Heinrich, L. (1984). Nonuniform estimates and asymptotic expansions of the remainder in the central limit theorem for \(m\)-dependent random variables. Math. Nachr. 115 7–20. · Zbl 0558.60026 · doi:10.1002/mana.19841150102
[15] Ho, S.-T. and Chen, L. H. Y. (1978). An \(L_p\) bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231–249. · Zbl 0375.60028 · doi:10.1214/aop/1176995570
[16] Nagaev, S. V. (1965). Some limit theorems for large deviations. Theory Probab. Appl. 10 214–235. · Zbl 0144.18704 · doi:10.1137/1110027
[17] Petrov, V. V. (1995). Limit Theorems of Probability Theory : Sequences of Independent Random Varaibles. Clarendon Press, Oxford. · Zbl 0826.60001
[18] Prakasa Rao, B. L. S. (1981). A nonuniform estimate of the rate of convergence in the central limit theorem for \(m\)-dependent random fields. Z. Wahrsch. Verw. Gebiete 58 247–256. · Zbl 0465.60028 · doi:10.1007/BF00531565
[19] Rinott, Y. (1994). On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55 135–143. · Zbl 0821.60037 · doi:10.1016/0377-0427(94)90016-7
[20] Rinott, Y. and Rotar, V. (1996). A multivariate CLT for local dependence with \(n^-1/2\log n\) rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333–350. · Zbl 0859.60019 · doi:10.1006/jmva.1996.0017
[21] Shergin, V. V. (1979). On the convergence rate in the central limit theorem for \(m\)-dependent random variables. Theory Probab. Appl. 24 782–796. · Zbl 0447.60023 · doi:10.1137/1124090
[22] Stein, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 583–602. Univ. California Press, Berkeley. · Zbl 0278.60026
[23] Stein, C. (1986). Approximation Computation of Expectations. IMS, Hayward, CA. · Zbl 0721.60016
[24] Sunklodas, J. (1999). A lower bound for the rate of convergence in the central limit theorem for \(m\)-dependent random fields. Theory Probab. Appl. 43 162–169. · Zbl 0938.60021 · doi:10.1137/S0040585X97976787
[25] Tihomirov, A. N. (1980). Convergence rate in the central limit theorem for weakly dependent random variables. Theory Probab. Appl. 25 800–818. · Zbl 0448.60019
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.