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