A note on convergence in mean for $d$-dimensional arrays of random vectors in Hilbert spaces under the Ces\`{a}ro uniform integrability

Abstract

This note establishes convergence in mean of order $p$, $0<p\le 1$ for $d$-dimensional arrays of random vectors in Hilbert spaces under the Ces\`{a}ro uniform integrability conditions. In the case where $0<p<1$, our $L_p$ convergence is valid irrespective of any dependence structure. In the case where $p=1$, the underlying random vectors are supposed to be pairwise independent. The mean convergence results are established for maximal partial sums while previous contributions were so far considered partial sums only. Some results in the literature are extended. Various properties of the Ces\`{a}ro uniform integrability of $d$-dimensional arrays of random vectors such as the classical equivalent criterion and the de La Vall\'{e}e Poussin theorem are also detailed.