Author:
S. Hörmann Graz University of Technology Institute of Statistics Steyrergasse 17/IV 8010 Graz Austria

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## Abstract

Let (Xk) be a sequence of independent r.v.’s such that for some measurable functions gk : RkR a weak limit theorem of the form

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$g_k (X_1 , \ldots ,X_k )\xrightarrow{\mathcal{L}}G$$ \end{document}
holds with some distribution function G. By a general result of Berkes and Csáki (“universal ASCLT”), under mild technical conditions the strong analogue
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\frac{1} {{D_N }}\sum\limits_{k = 1}^N {d_k I\left\{ {g_k (X_1 , \ldots ,X_k ) \leqq x} \right\} \to G(x)} a.s.$$ \end{document}
is also valid, where (dk) is a logarithmic weight sequence and DN = ∑k=1Ndk. In this paper we extend the last result for a very large class of weight sequences (dk), leading to considerably sharper results. We show that logarithmic weights, used traditionally in a.s. central limit theory, are far from optimal and the theory remains valid with averaging procedures much closer to, in some cases even identical with, ordinary averages.

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Acta Mathematica Hungarica
Language English
Size B5
Year of
Foundation
1950
Volumes
per Year
3
Issues
per Year
6
Founder's
H-1051 Budapest, Hungary, Széchenyi István tér 9.
Springer Nature Switzerland AG
Publisher's
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
ISSN 0236-5294 (Print)
ISSN 1588-2632 (Online)

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