Abstract
The weighted averages of a sequence (c k ), c k ∈ ℂ, with respect to the weights (p k ), p k ≥ 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 σ n → L as n → ∞ or σ(t) → L as t → ∞ exists, respectively. These characterizations may find applications in probability theory.
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References
I. Berkes, E. Csáki and L. Horváth, An almost sure central limit theorem under minimal conditions, Statist. Probab. Lett., 37 (1998), 67–76.
G. H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949.
F. MóRICZ, On the harmonic averages of numerical sequences, Arch. Math. (Basel), 86 (2006), 375–384.
F. MóRICZ, U. Stadtmüller and M. Thalmeier, Strong laws for blockwise M-dependent random fields, J. Theoret. Probab., 21 (2008), 660–671.
P. RéVéSZ, The Laws of Large Numbers, Akadémiai Kiadó, Budapest, 1967.
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Communicated by István Berkes
Supported by the TAMOP-4.2.1/B-09/1/KONV-2010-0005 project.
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Móricz, F., Stadtmüller, U. Characterization of the convergence of weighted averages of sequences and functions. Period Math Hung 65, 135–145 (2012). https://doi.org/10.1007/s10998-012-9329-4
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DOI: https://doi.org/10.1007/s10998-012-9329-4