Mean convergence for Banach space-valued random elements indexed in measure spaces
Abstract
This article studies mean convergence of Banach space-valued random elements indexed in a family of finite measure spaces. We derive Lp-convergence theorems under (compact) uniform integrability in two regimes: a decaying-index-mass regime and a bounded-index-mass regime, the latter requiring a new dependence structure which is called diagonal negative dependence for the random elements and expressed via the self-product of the index measure. We provide examples showing that the conditions to obtain the results are sharp and strictly weaker than related conditions in the literature. As a further illustration for the index measure space framework, a functional law of large numbers in Lp on the space of continuous functions is derived, where the random elements are solutions of stochastic differential equations driven by Brownian motions extracted from a common Brownian sheet.
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BibTeX
@misc{sang2026mean,
title = {{Mean convergence for Banach space-valued random elements indexed in measure spaces}},
author = {Nguyen Thi Kim Sang and Nguyen Tran Thuan},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30357},
eprint = {2608.30357},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30357},
}
RIS
TY - UNPB
TI - Mean convergence for Banach space-valued random elements indexed in measure spaces
AU - Nguyen Thi Kim Sang
AU - Nguyen Tran Thuan
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30357
UR - https://arxiv.org/abs/2608.30357
AB - This article studies mean convergence of Banach space-valued random elements indexed in a family of finite measure spaces. We derive L^p-convergence theorems under (compact) uniform integrability in two regimes: a decaying-index-mass regime and a bounded-index-mass regime, the latter requiring a new dependence structure which is called diagonal negative dependence for the random elements and expressed via the self-product of the index measure. We provide examples showing that the conditions to obtain the results are sharp and strictly weaker than related conditions in the literature. As a further illustration for the index measure space framework, a functional law of large numbers in L^p on the space of continuous functions is derived, where the random elements are solutions of stochastic differential equations driven by Brownian motions extracted from a common Brownian sheet.
ER -