发表机构
Vinh University; University of Economics Ho Chi Minh City（荣市大学; 胡志明市经济大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究以有限测度空间为索引的巴拿赫空间值随机元的均值收敛，在两类索引质量区域推导$L^p$收敛定理，提出对角负依赖结构，推导连续函数空间上的泛函大数定律，所得条件弱于文献相关条件。
AI 中文摘要
本文研究以一族有限测度空间为索引的巴拿赫空间值随机元的均值收敛。我们在（紧）一致可积性条件下，针对两类索引质量区域推导了$L^p$收敛定理：衰减索引质量区域与有界索引质量区域，后者要求一种新的依赖结构，称为随机元的对角负依赖，通过索引测度的自积来表达。我们提供例子表明，获得该结果的条件是尖锐的，且严格弱于文献中相关条件。作为索引测度空间框架的进一步说明，我们推导了连续函数空间上$L^p$中的泛函大数定律，其中随机元是由公共布朗片导出的布朗运动驱动的随机微分方程的解。
英文摘要
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.
