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Multivariate normal approximation with Stein’s method of exchangeable pairs under a general linearity condition. (English) Zbl 1200.62010
Stein’s method has been suggested to asses the distance between univariate random variables and normal distributions. It makes use of the method of exchangeable pairs [see C. Stein, Inst. Math. Stat. Lect. Notes - Monogr. Ser. 7 (1986; Zbl 0721.60016); A. Röllin, Stat. Probab. Lett. 78, No. 13, 1800–1806 (2008; Zbl 1147.62017); and S. Chatterjee and E. Meckes, ALEA, Lat. Am. J. Probab. Math. Stat. 4, 257–283, electronic only (2008; Zbl 1162.60310)]. The paper establishes a multivariate exchangeable pairs approach enabling to asses distances to potentially singular multivariate normal distributions. An embedding method is suggested allowing for a normal approximation even when the corresponding statistics are inappropriate for Stein’s exchangeable pairs approach. Examples of runs on the line and double-indexed permutation statistics illustrate the method.

MSC:
62E17 Approximations to statistical distributions (nonasymptotic)
62H05 Characterization and structure theory for multivariate probability distributions; copulas
60F05 Central limit and other weak theorems
60C05 Combinatorial probability
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