Convergence in the \(p\)th-mean and some weak laws of large numbers for weighted sums of random elements in separable normed linear spaces (Q797889)

Let \(X_ n\), \(n\geq 1\), be a sequence of random elements taking values in a separable Banach space B and \(a_{nk}\), \(n\geq 1\), \(k\geq 1\), a double array of real numbers satisfying \(\sum_{k}| a_{nk}|\leq \Gamma\) for every \(n\geq 1\) for some positive constant \(\Gamma\). The main result of the paper under review can be described as follows. If \(\| X_ n\|\), \(n\geq 1\), is uniformly integrable, then \(\sum_{k}a_{nk}X_ k\), \(n\geq 1\), converges to 0 in probability if and only if \(\sum_{k}a_{nk}f(X_ k)\), \(n\geq 1\), converges to 0 in probability for every f in the dual space \(B^*\).