Weak laws of large numbers for double arrays of random elements in Banach spaces (Q2431301)

Let \((V_{mn})\) be random elements in a martingale type \(p\) Banach space. The author provides conditions for \[ \max_{1\leq k\leq u_m,\,1\leq l\leq v_n}\frac{1}{a_{mn}} \left\|\sum_{i=1}^{k}\sum_{j=1}^{l}(V_{ij}- c_{mnij})\right\| \to 0 \] and \[ a_{mn}^{-1}\sum_{i=1}^{T_m} \sum_{j=1}^{\tau_n}(V_{ij}-c_{mnij})\to 0 \] in probability as \(m,n\to\infty\) where: \((u_m)\), \((v_n)\), \((a_{mn})\) and \((b_{mn})\) are positive integers; \((T_m)\) and \((\tau_n)\) are positive integer valued random variables; \(\mathcal{F}_{ij}=\sigma(V_{kl})_{k<i}^{l<j}\); and \(c_{mnij}=E(V_{ij}I(\|V_{ij}\|\leq b_{mn}|\mathcal{F}_{ij}))\).