Niemiro, Wojciech; Zieliński, Ryszard Remarks on uniform convergence of random variables and statistics. (English) Zbl 1269.62030 Demonstr. Math. 45, No. 2, 265-274 (2012). Summary: Convergence in distribution, convergece in probability, and convergence almost surely, uniform with respect to a family of probability distributions, are considered. These concepts appeared to be appropriate tools for asymptotic theory of mathematical statistics and many partial results are scattered in the literature of this subject. The aim of this note is to present a unified review of the results in a general and abstract setup. We examine a few rather paradoxical examples which hopefully shed some light on the subtleties of the underlying definitions and the role of asymptotic approximations in statistics. A motivation for considering these problems is provided by their applications. MSC: 62E20 Asymptotic distribution theory in statistics 60F05 Central limit and other weak theorems 60F15 Strong limit theorems PDFBibTeX XMLCite \textit{W. Niemiro} and \textit{R. Zieliński}, Demonstr. Math. 45, No. 2, 265--274 (2012; Zbl 1269.62030) Full Text: DOI References: [1] J. R. Berrendero, R. H. Zamar, A note on the uniform asymptotic normality of location M-estimates, Metrika 63 (2006), 55-69.; · Zbl 1095.62027 [2] A. A. Borovkov, Mathematical Statistics, Gordon and Breach (1998).; · Zbl 0913.62002 [3] Z. Ciesielski, R. Zieliński, Polynomial and spline estimators of the distribution function with prescribed accuracy, Applicationes Mathematicae 36(1) (2009), 1-12.; · Zbl 1284.62249 [4] R. M. Dudley, Real Analysis and Probability, Wadsworth & Brooks (1989).; · Zbl 0686.60001 [5] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II, Wiley (1966).; · Zbl 0138.10207 [6] I. A. Ibragimov, R. Z. Khas’minskii, Asymptotic theory of estimation (in Russian), „Nauka”, Moscow, 1979 [MR545339, 81h:62004].; [7] W Niemiro, R. Zieliński, Uniform asymptotic normality for the Bernoulli scheme, Appl. Math. 34(2) (2007), 215-221.; · Zbl 1121.60019 [8] M Omelka, M. Salibian-Barrera, Uniform asymptotics for S- and MM-regression estimators, Ann. Inst. Stat. Math. 62 (2010), 897-927.; · Zbl 1432.62210 [9] M. Salibian-Barrera, R. H. Zamar, Uniform asymptotics for robust location estimates when the scale is unknown, Ann. Statist. 32(4) (2004), 1434-1447.; · Zbl 1047.62022 [10] R. Zieliński, Uniform strong consistency of sample quantiles, Statist. Probab. Lett. 37 (1998), 115-119.; · Zbl 0891.62035 [11] R. Zieliński, Effective WLLN, SLLN and CLT in statistical models, Appl. Math. 31(1) (2004), 117-125.; · Zbl 1051.62019 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.