We revisit the question of whether the strong law of large numbers (SLLN) holds uniformly in a rich family of distributions, culminating in a distribution-uniform generalization of the Marcinkiewicz-Zygmund SLLN. These results can be viewed as extensions of Chung's distribution-uniform SLLN to random variables with uniformly integrable $q^\text{th}$ absolute central moments for $0 < q < 2;\ q \neq 1$. Furthermore, we show that uniform integrability of the $q^\text{th}$ moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of $n^{1/q - 1}$. These proofs centrally rely on distribution-uniform analogues of some familiar almost sure convergence results including the Khintchine-Kolmogorov convergence theorem, Kolmogorov's three-series theorem, a stochastic generalization of Kronecker's lemma, and the Borel-Cantelli lemmas. The non-identically distributed case is also considered.

翻译：我们重新审视了强大数定律（SLLN）是否在丰富族分布中一致成立的问题，最终得到了Marcinkiewicz-Zygmund SLLN的分布均匀推广。这些结果可视为Chung分布均匀SLLN向具有一致可积的$q^\text{th}$阶绝对中心矩（其中$0 < q < 2$且$q \neq 1$）的随机变量族的扩展。进一步地，我们证明了$q^\text{th}$阶矩的一致可积性是SLLN以Marcinkiewicz-Zygmund速率$n^{1/q - 1}$一致成立的充分必要条件。这些证明的核心依赖于一些经典几乎必然收敛结果的分布均匀类比，包括Khintchine-Kolmogorov收敛定理、Kolmogorov三级数定理、Kronecker引理的随机推广以及Borel-Cantelli引理。此外，本文还考虑了非同分布情形。