Bolthausen, Erwin An estimate of the remainder in a combinatorial central limit theorem. (English) Zbl 0563.60026 Z. Wahrscheinlichkeitstheor. Verw. Geb. 66, 379-386 (1984). The author describes a surprisingly short inductive version of the so- called Stein method to prove the Berry-Esseen theorem for i.i.d. random variables. The same methods (with some modifications) are applied to prove the Berry-Esseen theorem for general linear rank statistics (of type \(\sum^{n}_{i=1}a_{i\pi (i)}\), \(\pi\) random permutation) under optimal conditions. This clever inductive approach yields a much shorter proof and more insight into the minimal conditions as previous Fourier- theoretic approaches as well as previous applications of Stein’s method to this problem. Reviewer: F.Götze Cited in 5 ReviewsCited in 52 Documents MSC: 60F05 Central limit and other weak theorems 62E20 Asymptotic distribution theory in statistics Keywords:Stein method; Berry-Esseen theorem; rank statistics × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Does, R. J.M. M., Berry-Esseen theorems for simple linear rank statistics, Ann. Probability, 10, 982-991 (1982) · Zbl 0498.62020 · doi:10.1214/aop/1176993719 [2] Ho, S. T.; Chen, L. H.Y., An L_p bound for the remainder in a combinatorial central limit theorem, Ann. Probability, 6, 231-249 (1978) · Zbl 0375.60028 · doi:10.1214/aop/1176995570 [3] Hoeffding, W., A combinatorial central limit theorem, Ann. Math. Statist., 22, 558-566 (1951) · Zbl 0044.13702 · doi:10.1214/aoms/1177729545 [4] Husková, Marie, The Berry-Esseen theorem for rank statistics, Comment. Math. Univ. Carolina, 20, 399-415 (1979) · Zbl 0416.60023 [5] Motoo, M., On the Hoeffding’s combinatorial central limit theorem, Ann. Inst. Statist. Math., 8, 145-154 (1957) · Zbl 0084.13802 · doi:10.1007/BF02863580 [6] Stein, Ch., A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Sympos. Math. Statist. Probability, 2, 583-602 (1972) · Zbl 0278.60026 [7] von Bahr, B., Remainder term estimate in a combinatorial limit theorem, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 35, 131-139 (1976) · Zbl 0366.60028 · doi:10.1007/BF00533317 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.