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On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein’s inequality conditions. (English) Zbl 0612.60031
Let F be the distribution function of a sum of independent k-dimensional random vectors, and let \(\Phi\) be the Gaussian distribution with zero mean and the same covariance operator as F. The main results consider the closeness of F and \(\Phi\) ; in particular, an upper bound on the Lévy- Prohorov distance between F and \(\Phi\) is presented, and shown to be optimal, in the case of infinitely divisible distributions and certain distributions with exponentially-decreasing tails.
The proofs are based on techniques used by V. V. Yurinskij [Teor. Veroyatn. Primen. 22, 675-688 (1977; Zbl 0399.60022)]. Similar approximation results are given for density functions and conditional distributions.
Reviewer: R.J.Tomkins

MSC:
60F10 Large deviations
60E07 Infinitely divisible distributions; stable distributions
60F99 Limit theorems in probability theory
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