Acta Mathematica Hungarica
Volume/Issue:
Volume 116: Issue 4
On the universal A.S. central limit theorem
Author: S. Hörmann 1
View More View Less
  • 1 Graz University of Technology Institute of Statistics Steyrergasse 17/IV 8010 Graz Austria
DOI:
https://doi.org/10.1007/s10474-007-6070-1
Page Count:
377–398
Publication Date:
01 Sep 2007
Online Publication Date:
26 May 2007
Article Category:
Research Article
Restricted access

Abstract  

Let (Xk) be a sequence of independent r.v.’s such that for some measurable functions gk : Rk → R 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.

Cover Acta Mathematica Hungarica
Acta Mathematica Hungarica
Print ISSN:
0236-5294
Online ISSN:
1588-2632
Keywords:
almost sure limit theory; summation methods; 60F15; 60F05

Monthly Content Usage

Abstract Views Full Text Views PDF Downloads
Sep 2020 0 0 0
Oct 2020 0 0 0
Nov 2020 0 0 0
Dec 2020 0 0 0
Jan 2021 0 0 0
Feb 2021 0 0 0
Mar 2021 0 0 0