On the concept of B-statistical uniform integrability of weighted sums of random variables and the law of large numbers with mean convergence in the statistical sense

In this correspondence, for a nonnegative regular summability matrix B and an array {ank} of real numbers, the concept of B-statistical uniform integrability of a sequence of random variables {Xk} with respect to {ank} is introduced. This concept is more general and weaker than the concept of {Xk} being uniformly integrable with respect to {ank}. Two characterizations of B-statistical uniform integrability with respect to {ank} are established, one of which is a de La Vallee Poussin-type characterization. For a sequence of pairwise independent random variables {Xk} which is B-statistically uniformly integrable with respect to {ank}, a law of large numbers with mean convergence in the statistical sense is presented for 8 k=1 ank( Xk - EXk) as n. 8. A version is obtained without the pairwise independence assumption by strengthening other conditions.