Studia Scientiarum Mathematicarum Hungarica
Volume/Issue:
Volume 50: Issue 1
Characterization of the convergence of weighted averages in a more general setting
Authors: Ferenc Móricz 1 and Ulrich Stadtmüller 2
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  • 1 University of Szeged Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
  • 2 Universität Ulm Abt. Zahlentheorie und Wahrscheinlichkeitstheorie D-89069 Ulm Germany
DOI:
https://doi.org/10.1556/sscmath.50.2013.1.1230
Page Count:
51–66
Publication Date:
01 Mar 2013
Online Publication Date:
26 Apr 2013
Article Category:
Research Article
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Let ν be a positive Borel measure on ℝ̄+:= [0;∞) and let p: ℝ̄+ → ℝ̄+ be a weight function which is locally integrable with respect to ν. We assume that \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} $P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$ \end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \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} $\sigma _t (f;pd\nu ): = \frac{1}{{P(t)}}\int\limits_0^t {f(u)p(u)d\nu (u)} $ \end{document} for large enough t, where P(t) > 0. We prove necessary and sufficient conditions under which the finite limit \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} $\sigma _t (f;pd\nu ) \to Last \to \infty $ \end{document} exists. This characterization is a unified extension of the results in [5], and it may find application in Probability Theory and Stochastic Processes.

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Cover Studia Scientiarum Mathematicarum Hungarica
Studia Scientiarum Mathematicarum Hungarica
Print ISSN:
0081-6906
Online ISSN:
1588-2896
Keywords:
Primary 40A10; Secondary 26D15, 28A26, 60F15, 60H99; Weighted averages of a locally integrable function f: ℝ̄+ → ℂ; Characterization of convergence in the mean; positive Borel measure ν on ℝ̄+; weight function p: ℝ̄+ → ℝ̄+; weighted averages of a locally integrable function f: ℝ̄+ → ℂ with respect to p dν; generalized inverse function of a nondecreasing function; Strong Law of Large Numbers for Stochastic Processes

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