Mean Convergence Theorems for the Maximum Normed Partial Sums of Random Vectors in Hilbert Spaces under a General Condition of Weighted Integrability

This paper establishes mean convergence for the maximum normed partial sums of \(\mathcal{H}\) -valued random vectors under a general condition of weighted integrability. Our \(L_{p}\) convergence holds regardless of any dependence structure when \(0<p<1\) . For \(p=1\) , the main theorem is applicable to arrays of random vectors exhibiting (i) pairwise and coordinatewise negative dependence, (ii) coordinatewise negative association, and (iii) coordinatewise widely orthant dependence. Notably, the results for cases (i) and (iii) build upon and enhance the main theorem presented by Cabrera and Volodin [J. Math. Anal. Appl. 305, 664–658 (2005)] and Wu et al. [Stochastics 91, 916–944 (2019)], respectively.