The weak law of large numbers in martingale type \(p\) Banach spces under a general condition of Cesàro type (Q1589195)

Let \(\{V_{nj}, j\geq 1, n\geq 1\}\) be an array of \(B\)-valued random elements, where \(B\) is a Banach space of martingale type \(p\), \(1\leq p\leq 2\). Let \(\{a_{nj}, j\geq 1, n\geq 1\}\) be an array of (nonrandom) constants. Let us put \(Z_n= \sum^{N_n}_{j=1} a_{nj}(V_{nj}- E(V^*_{nj}\mid F_{nj- 1}))\), where \(V^*_{nj}= V_{nj}I(\|V_{nj}\|\leq g(k_n))\), \(F_{nj}= \sigma(V_{ni}, 1\leq i\leq j)\), \(j\geq 1\), \(n\geq 1\), \(F_{n0}= \{\emptyset,\Omega\}\), \(n\geq 1\), \(\{N_n, n\geq 1\}\) is a sequence of positive integer-valued random variables and \(g(k_n)\), \(n\geq 1\), is a sequence of positive numbers satisfying some additional conditions. The authors present sufficient conditions under which \(Z_n\to 0\), in probability, as \(n\to \infty\).