Weighted Laws of Large Numbers for a class of independent summands

Abstract

This paper obtains a necessary and sufficient condition for a weak law of large numbers for weighted averages of positive-valued independent random variables whose distributions belong to a class which includes the F-scheme of record theory. Additional general conditions are found under which the weak law extends to a strong law with the same norming. Examples show these conditions can be fulfilled, and that if they are not, then the weighted averages exhibit multiple growth rates. 2020 Mathematics Subject Classification: Primary 60F05, 60F15; Secondary 60E10.