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 $0 < q < 2$. 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 novel 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. We also consider the non-identically distributed case.

翻译：我们重新审视了强大数定律（SLLN）是否在丰富的分布族中一致成立的问题，最终得到了Marcinkiewicz-Zygmund强大数定律的一个分布一致推广。这些结果可视为Chung的分布一致强大数定律向具有一致可积$q$阶绝对中心矩（$0 < q < 2$）的随机变量的扩展。此外，我们证明了$q$阶矩的一致可积性是强大数定律以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对若干常见几乎必然收敛结果的创新性分布一致类比，包括Khinchine-Kolmogorov收敛定理、Kolmogorov三级数定理、Kronecker引理的一个随机推广，以及Borel-Cantelli引理。我们也考虑了非同分布的情形。