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  • 1 University of Szeged Bolyai Institute Aradi vértanúk tere 1 H-6720 Szeged Hungary
  • 2 University of North Carolina Department of Statistics and Operations Research 210 Smith Building Chapel Hill NC 27599-3260 USA
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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.