On the weak law of large numbers for weighted sums of pairwise negative quadrant dependent random variables (Q2707986)

Let \(\{X_j:j\geq 1\}\) be a sequence of pairwise negatively quadrant dependent r.v.'s [\(P(X_i\leq x, X_j\leq y) \leq P(X_i\leq x)P(X_j\leq y)\); see \textit{E. L. Lehmann}, Ann. Math. Stat. 37, 1137-1153 (1966; Zbl 0146.40601)] such that the tail functions of their distributions are dominated by the one associated to a certain r.v. Let \(\{a_j\}, \{b_j\}\) be sequences of positive constants, \(b_j\rightarrow\infty\). Sufficient conditions are given in order that \(b_n^{-1}\sum_{j=1}^n a_j(X_j-c_{nj})\) tends to zero in probability for suitable constants \(c_{nj}\).