Normal approximation under local dependence

Annals of Probability

Volumn 32, Issue 3 A, 2004, Pages 1985-2028

  Chen, Louis H Y ^a   Shao, Qi Man ^a,b  

^a NATIONAL UNIVERSITY OF SINGAPORE   (Singapore)

^b UNIVERSITY OF OREGON   (United States)

Author keywords

Concentration inequality; Local dependence; Nonuniform Berry Esseen bound; Normal approximation; Random field; Stein's method; Uniform Berry Esseen bound

Indexed keywords

EID: 4544375370     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117904000000450     Document Type: Article

References (25)

1. BALDI, P. and RINOTT, Y. (1989). On normal approximation of distributions in terms of dependency graph. Ann. Probab. 17 1646-1650.
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.
3. BERRY, A. C. (1941). The accuracy of the Gaussian approximation to the sum of independent variables. Trans. Amer. Math. Soc. 49 122-136.
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.
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.
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.
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.
9. DASGUPTA, R. (1992). Nonuniform speed of convergence to normality for some stationary m-dependent processes. Calcutta Statist. Assoc. Bull. 42 149-162.
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.
11. 1 bounds for asymptotic normality of m-dependent sums using Stein's technique. Ann. Probab. 2 522-529.
12. ESSEEN, C. G. (1945). Fourier analysis of distribution functions: A mathematical study of the Laplace-Gaussian law. Acta Math. 77 1-125.
13. ESSEEN, C. G. (1968). On the concentration function of a sum of independent random variables. Z. Wahrsch. Verw. Gebiete 9 290-308.
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.
15. p bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231-249.
16. NAGAEV, S. V. (1965). Some limit theorems for large deviations. Theory Probab. Appl. 10 214-235.
17. PETROV, V. V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Varaibles. Clarendon Press, Oxford.
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.
19. RINOTT, Y. (1994). On normal approximation rates for certain sums of dependent random variables. J. Comput. Appl. Math. 55 135-143.
20. -1/2 log n rate and applications to multivariate graph related statistics. J. Multivariate Anal. 56 333-350.
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.
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.
23. STEIN, C. (1986). Approximation Computation of Expectations. IMS, Hayward, CA.
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.
25. TIHOMIROV, A. N. (1980). Convergence rate in the central limit theorem for weakly dependent random variables. Theory Probab. Appl. 25 800-818.