Koshevnik, Yu. A. On some limit properties and a method of construction of asymptotically optimal nonparametric estimations of distribution functions. (Russian) Zbl 0574.62039 Teor. Veroyatn. Primen. 29, No. 4, 772-778 (1984). 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 Cited in 2 ReviewsCited in 1 Document MSC: 62G05 Nonparametric estimation 60F05 Central limit and other weak theorems Keywords:review of earlier results; uniform weak convergence; continuous transformations; linear complete metric space; integral kernel transformations; Brownian bridge; locally uniform central limit theorem PDFBibTeX XMLCite \textit{Yu. A. Koshevnik}, Teor. Veroyatn. Primen. 29, No. 4, 772--778 (1984; Zbl 0574.62039)