Convergence rate in CLT for vector-valued random fields with self-normalization

Statistical version of the central limit theorem (CLT) with random matrix normalization is established for random fields with values in a space Rk (k ≥ 1). Dependence structure of the field under consideration is described in terms of the covariance inequalities for the class of bounded Lipschitz ”test functions” defined on finite disjoint collections of random vectors constituting the field. The main result provides an estimate of the convergence rate, over a family of convex bounded sets, in the CLT with random normalization.

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To the memory of Professor Kazimierz Urbanik.