Some strong laws of large numbers for blockwise martingale difference sequences in martingale type \(p\) Banach spaces (Q1757963)

The paper consists of four parts, i.e., Introduction; Preliminaries; Main results; and, finally, Corollaries and examples/counterexamples. The authors study generalizations of well-known results by \textit{F. Móricz} [Proc. Am. Math. Soc. 101, 709--715 (1987; Zbl 0632.60025)] and \textit{V. F. Gaposhkin} [Sov. Math. 34, No. 5, 13--20 (1990); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1990, No. 5(336), 12--18 (1990; Zbl 0716.42024); Theory Probab. Appl. 39, No. 4, 677--684 (1994); translation from Teor. Veroyatn. Primen. 39, No. 4, 804--812 (1994; Zbl 0847.60022)] and others in the following manner. For a blockwise martingale difference sequence of random elements \(\{V_n,n\geq 1\}\) taking values in a real separable martingale type \(p\) Banach space \((1\leq p\leq 2)\), conditions are provided for strong laws of large numbers of the form \[ \lim_{n\to\infty} \sum^n_{i=1} V_i/g_n= 0 \] almost surely to hold where the constants \(g_n\uparrow\infty\). Additionally, a result by \textit{P. Hall} and \textit{C. C. Heyde} [Martingale limit theory and its application. New York etc.: Academic Press, A Subsidiary of Harcourt Brace Jovanovich, Publishers (1980; Zbl 0462.60045)] which was obtained for sequences of random variables is extended to the setting of a martingale type \(p\) Banach space \((1< p\leq 2)\) and to hold with a Marcinkiewicz-Zygmund type normalization. Illustrative examples and counterexamples are provided.