On the laws of large numbers for blockwise martingale differences and blockwise adapted sequences (Q817056)

Let \(\{\omega(n):n\geq 1\}\) be a strictly increasing sequence of positive integers with \(\omega(1)=1\). For each \(k\geq 1\), we set \(\Delta_k=[\omega(k), \omega(k+1))\). Let \(\{{\mathcal F}_i:i\geq 1\}\) be a sequence of \(\sigma\)-fields such that for each fixed \(k\), the sequence \(\{{\mathcal F}_i:i\in\Delta_k\}\) is increasing. A sequence \(\{X_i:i\geq 1\}\) of random variables is said to be blockwise adapted to \(\{{\mathcal F}_i:i\geq 1\}\) if each \(X_i\) is measurable with respect to \({\mathcal F}_i\). The sequence \(\{X_i,{\mathcal F}_i:i\geq 1\}\) is called a blockwise martingale difference with respect to the blocks \(\{\Delta_k:k\geq 1\}\) if for each fixed \(k\), the sequence \(\{X_i,{\mathcal F}_i:i\in\Delta_k\}\) is a martingale difference. Two theorems and four corrollaries are proved on the laws of large numbers indicated in the title. Some known results from the literature are extended in this way.