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On some limit properties and a method of construction of asymptotically optimal nonparametric estimations of distribution functions. (Russian) Zbl 0574.62039

The paper is a review of earlier results which were published in research reports. At first four theorems are presented. These theorems give sufficient conditions for the uniform weak convergence of a sequence of random elements and their continuous transformations. These random elements are functions from a linear complete metric space.
Let us have a random sample from the distribution P. Let \(\Phi\) (P), \(\Phi_ n\) and \(V_ p\) denote integral kernel transformations of the distribution P, the sample and the Brownian bridge related to P, respectively. In this paper the property of uniform weak convergence \[ \sqrt{n}[\Phi_ n-\Phi (P)]\to^{{\mathcal D}}V_ p \] is called ”a locally uniform central limit theorem” for \(\Phi_ n.\)
In the second part of the paper the author presents sufficient conditions for the locally uniform central limit theorem for some \(\Phi_ n\) and their transforms. Several examples of application of the theorems to particular \(\Phi_ n's\) are given.
Reviewer: P.Staniewski

MSC:

62G05 Nonparametric estimation
60F05 Central limit and other weak theorems
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