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  • 1 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
  • 2 Abt. Zahlentheorie und Wahrscheinlichkeitstheorie, Universität Ulm, D-89069 Ulm, Germany
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Abstract  

The weighted averages of a sequence (ck), ck ∈ ℂ, with respect to the weights (pk), pk ≥ 0, with {fx135-1} are defined by {fx135-2} while the weighted average of a measurable function f: ℝ+ → ℂ with respect to the weight function p(t) ≥ 0 with {fx135-3}. Under mild assumptions on the weights, we give necessary and sufficient conditions under which the finite limit σnL as n → ∞ or σ(t) → L as t → ∞ exists, respectively. These characterizations may find applications in probability theory.