Abstract
Based on a stochastic extension of Karamata’s theory of slowly varying functions, necessary and sufficient conditions are established for weak laws of large numbers for arbitrary linear combinations of independent and identically distributed nonnegative random variables. The class of applicable distributions, herein described, extends beyond that for sample means, but even for sample means our theory offers new results concerning the characterization of explicit norming sequences. The general form of the latter characterization for linear combinations also yields a surprising new result in the theory of slow variation.
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Communicated by István Berkes
Work was supported in part by the Hungarian Scientific Research Fund, Grant T048360, and carried out within the Analysis and Stochastics Research Group of the Hungarian Academy of Sciences.
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Csörgő, S., Simons, G. Weak laws of large numbers for cooperative gamblers. Period Math Hung 57, 31–60 (2008). https://doi.org/10.1007/s10998-008-7031-z
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DOI: https://doi.org/10.1007/s10998-008-7031-z