On the weak law of large numbers for arrays (Q1344832)

A weak law of large numbers is proved for an array \(X_{ni}\), \(1 \leq i \leq k_ n\), \(n \geq 1\), \(k_ n \to \infty\), under the assumption that for some \(p \in (0,2)\), \[ \lim_{a \to \infty} {1\over k_ n} \sum^{k_ n}_{i = 1} aP(| X_{ni}| > a) = 0 \text{ uniformly in }n. \] Compare \textit{A. Gut's} result [ibid. 14, No. 1, 49-52 (1992; Zbl 0769.60034)] who obtained \(L^ p\)-convergence on a somewhat stronger domination condition.