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 $$\left\{ a_{nk}\right\} $$ a nk of real numbers, the concept of B-statistical uniform integrability of a sequence of random variables $$\left\{ X_{k}\right\} $$ X k with respect to $$\left\{ a_{nk}\right\} $$ a nk is introduced. This concept is more general and weaker than the concept of $$\left\{ X_{k}\right\} $$ X k being uniformly integrable with respect to $$\left\{ a_{nk}\right\} $$ a nk . Two characterizations of B-statistical uniform integrability with respect to $$\left\{ a_{nk}\right\} $$ a nk are established, one of which is a de La Vallée Poussin-type characterization. For a sequence of pairwise independent random variables $$\left\{ X_{k}\right\} $$ X k which is B-statistically uniformly integrable with respect to $$\left\{ a_{nk}\right\} $$ a nk , a law of large numbers with mean convergence in the statistical sense is presented for $$\sum \nolimits _{k=1}^{\infty }a_{nk}(X_{k}-\mathbb {E}X_{k})$$ ∑ k = 1 ∞ a nk ( X k - E X k ) as $$n\rightarrow \infty $$ n → ∞ . A version is obtained without the pairwise independence assumption by strengthening other conditions.