Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces

Maximal inequalities for normed double sums of random elements in martingale type <i>p</i> Banach spaces with applications to degenerate mean convergence of the maximum of normed sums

“…It plays an important role in proving Theorem 4.1. For the special case p = q, Dung et al [5], Quang and Huan [17], and Son et al [23] obtained a similar result. Theorem 3.3.…”

“…In [18], Rosalsky and Thành established a Kolmogorov-Doob-type maximal inequality for normed double sums of independent random elements in a Rademacher type p Banach space. Dung et al [5], Quang and Huan [17], and Son et al [23] established a Kolmogorov-Doob-type maximal inequality for normed double sums of random elements taking values in a martingale type p Banach space. In this paper, we further generalize the Dung et al [5], Quang and Huan [17], and Son et al [23] results by considering the case where the moments are of higher order than p. We then use the obtained result to obtain a mean convergence theorem for the maximum of normed and suitably centered double sums of random elements taking values in a real separable martingale type p Banach space.…”