Strong laws of large numbers for random fields in martingale type \(p\) Banach spaces
Let \(\mathbb{Z}^+_d\) be the coordinate-wise partially ordered positive integer lattice points. A typical point there is \({\mathbf n}= (n_1, n_2,\dots,n_d)\). Further, let \(S_n\), \({\mathbf n}\in\mathbb{Z}^+_d\) be a random walk taking values in a real separable Banach space of type \(p\in[1, 2]\). Finally, \(\alpha_k\), \(k= 1,2,\dots,d\), are positive reals. The main results concern Marcinkiewicz-Zygmund strong laws for the case when the coordinates are raised to different powers, viz., the normalization is \(\prod^d_{k=1} n^{\alpha_k}_k\). The particular case \(\alpha_k= 1/r\) for all \(k\) reduces the results to the standard form. If, in addition, \(d=1\), the original result from 1937 [\textit{J. Marcinkiewicz, A. Zygmund}, Fundam. Math. 29, 60--90 (1937; Zbl 0016.40901; JFM 63.0946.02)] is rediscovered.
scientific article; zbMATH DE number 5703978