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Multivariate normal approximation using exchangeable pairs. (English) Zbl 1162.60310
Summary: Since the introduction of Stein’s method in the early 1970s, much research has been done in extending and strengthening it; however, there does not exist a version of Stein’s original method of exchangeable pairs for multivariate normal approximation. The aim of this article is to fill this void. We present three abstract normal approximation theorems using exchangeable pairs in multivariate contexts, one for situations in which the underlying symmetries are discrete, and real and complex versions of a theorem for situations involving continuous symmetry groups. Our main applications are proofs of the approximate normality of rank $$k$$ projections of Haar measure on the orthogonal and unitary groups, when $$k=o(n)$$.

##### MSC:
 60F05 Central limit and other weak theorems 60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
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