Complete convergence in mean for double arrays of random variables with values 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.…”

“…Mean convergence and laws of large numbers for sums of Banach space-valued random elements have enjoyed a wide literature of investigation (see, e.g., Adler et al [1], Chen et al [2], Chen and Wang [3], Hoffman-Jørgensen and Pisier [6], Hu et al [7], Korzeniowski [10], Li et al [11], Ordóñez Cabrera [12], Ordóñez Cabrera et al [13], Parker and Rosalsky [14], Rosalsky and Thành [18], Rosalsky et al [20], Son et al [23], Thành and Yin [24], Wang and Rao [26], and the references therein). However, only a few of these investigations establish mean convergence for the maximum of normed sums which is of special interest.…”

“…The proof is based on ideas from Lemma 2.3 of Rosalsky and Thành [18] which establishes a maximal inequality for normed double sums of independent mean zero random elements in Rademacher type p Banach spaces. In the papers [18,23], the authors considered the p-th moment of the maximum partial sums. In some cases, it may be necessary to bound moments of order higher than p for either the partial sums or the maximum of the partial sums.…”

On complete convergence in mean for double sums of independent random elements in Banach spaces

“…(ii) Recently, Son, Thang and Dung [20] proved a result on complete convergence in mean of order p without assuming that the summands are independent. More precise, they proved that for arbitrary double array {V mn , m ≥ 1, n ≥ 1} in a real separable Banach space, the condition Their result and ours are not comparable and do not imply each other, and our proof is completely different from theirs.…”