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Some Mean Convergence Theorems for the Maximum of Normed Double Sums of Banach Space Valued Random Elements
Authors: Andrew Rosalsky, Lê Văn Thành and Nguyen Thi Thuy
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Acta Mathematica Sinica, English Series
: 40     : 1727–1740
: https://link.springer.com/article/10.1007/s10114-024-2669-1
: NCKH_1523_20241114153727.pdf
Publishing year: 7/2024
In this correspondence, we establish mean convergence theorems for the maximum of normed double sums of Banach space valued random elements. Most of the results pertain to random elements which are M-dependent. We expand and improve a number of particular cases in the literature on mean convergence of random elements in Banach spaces. One of the main contributions of the paper is to simplify and improve a recent result of Li, Presnell, and Rosalsky [Journal of Mathematical Inequalities, 16, 117–126 (2022)]. A new maximal inequality for double sums of M-dependent random elements is proved which may be of independent interest. The sharpness of the results is illustrated by four examples.
Double sum, mean convergence, Rademacher type p Banach space, Banach space valued random element, M-dependent random elements
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