Affiliation:
1. 1 University of Szeged Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
2. 2 Universität Ulm Abt. Zahlentheorie und Wahrscheinlichkeitstheorie D-89069 Ulm Germany
Abstract
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}
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$P(t): = \int\limits_0^t {p(u)d\nu (u) \to \infty } andP(t - 0)/P(t) \to 1ast \to \infty .$
\end{document} Let f: ℝ̄+ → ℂ be a locally integrable function with respect to p dν, and define its weighted averages by \documentclass{aastex}
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$\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}
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$\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.