L1 bounds in normal approximation

Annals of Probability

Volumn 35, Issue 5, 2007, Pages 1888-1930

  Goldstein, Larry ^a  

^a UNIVERSITY OF SOUTHERN CALIFORNIA   (United States)

Author keywords

Berry Esseen; Combinatorial CLT; Cone measure; Sampling; Stein's method

Indexed keywords

EID: 51549108701     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117906000001123     Document Type: Article

References (26)

1. ANTTILA, M., BALL, K. and PERISSINAKI, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723-4735. MR1997580
2. VON BAHR, B. (1976). Remainder term estimate in a combinatorial limit theorem. Z. Wahrsch. Verw. Gebiete 35 131-139. MR0418187
3. BARBOUR, A. D., HOLST, L. and JANSON, S. (1992). Poisson Approximation. Oxford Univ. Press. MR1163825
4. BICKEL, P. and DOKSUM, K. (1977). Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco. MR0443141
5. BOBKOV, S. (2003). On concentration of distributions of random weighted sums. Ann. Probab. 31 195-215. MR1959791
6. BOLTHAUSEN, E. (1984). An estimate of the reminder in a combinatorial central limit theorem. Z Wahrsch. Verw. Gebiete 66 379-386. MR0751577
7. DIACONIS, P. and FREEDMAN, D. (1987). A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaré Probab. Statist. 23 397-423. MR0898502
8. GOLDSTEIN, L. (2004). Normal approximation for hierarchical sequences Ann. Appl. Probab. 14 1950-1969. MR2099658
9. GOLDSTEIN, L. (2005). Berry-Esseen bounds for combinatorial central limit theorems and pattern occurrences, using zero and size biasing. J. Appl. Probab. 42 661-683. MR2157512
10. GOLDSTEIN, L. and REINERT, G. (1997). Stein's method and the zero bias transformation with application to simple random sampling Ann. Appl. Probab. 7935-952. MR1484792
11. GOLDSTEIN, L. and RINOTT, Y. (1996). On multivariate normal approximations by Stein's method and size bias couplings. J. Appl. Probab. 33 1-17. MR1371949
12. HOEFFDING, W. (1951). A combinatorial central limit theorem. Ann. Math. Statist. 22 558-566. MR0044058
13. HOEFFDING, W. (1955). The extrema of the expected value of a function of independent random variables. Ann. Math. Statist. 26 268-275. MR0070087
14. p bound for the remainder in a combinatorial central limit theorem. Ann. Probab. 6 231-249. MR0478291
15. LEFVRE, C. and UTEV, S. (2003). Exact norms of a Stein-type operator and associated stochastic orderings. Probab. Theory Related Fields 127 353-366. MR2018920
16. MECKES, M. and MECKES, E. (2007). The central limit problem for random vectors with symmetries. J. Theoret. Probab. To appear. Available at arXiv:math.PR/0505618.
17. MOTOO, M. (1957). On the Hoeffding's combinatorial central limit theorem. Ann. Inst. Statist. Math. Tokyo 8 .145-154. MR0089560
18. n. Ann. Inst. H Poincaré Probab. Statist. 39 241-261. MR1962135
19. RACHEV, S. T. (1984). The Monge-Kantorovich transference problem and its stochastic applications. Theory Probab. Appl. 29 647-676. MR0773434
20. n balls. Proc. Amer. Math. Soc. 110 217-224. MR1015684
21. STEIN, C. (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. In Proc. Sixth Berkeley Symp. Math. Statist. Probab. 2 583-602. Univ. California Press, Berkeley. MR0402873
22. STEIN, C. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135-1151. MR0630098
23. STEIN, C. (1986). Approximate Computation of Expectations. IMS, Hayward, CA. MR0882007
24. SUDAKOV, V. (1978). Typical distributions of linear functionals in finite-dimensional spaces of higher dimension. Soviet Math. Dokl. 19 1578-1582.
25. [Translated from Dokl. Akad. Nauk SSSR 243 (1978) 1402-1405.] MR0517198
26. WALD, A. and WOLFOWITZ, J. (1944). Statistical tests based on permutations of the observations. Ann. Math. Statist. 15 358-372. MR0011424