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

TL;DR

This paper proves convergence in mean for multi-dimensional arrays of Hilbert space-valued random vectors under Cesàro uniform integrability, extending previous results to maximal partial sums and different dependence structures.

Contribution

It establishes mean convergence of order p for arrays of random vectors in Hilbert spaces under Cesàro uniform integrability, including cases with dependence and independence.

Findings

Convergence in mean for maximal partial sums established.

Extension of classical criteria for Cesàro uniform integrability.

Results valid for arrays with dependence when 0<p<1.

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.