标题： 分布一致的大数定律 显示英文标题

标题： Distribution-uniform strong laws of large numbers

摘要： 我们重新审视强大数定律（SLLN）在丰富的分布族中是否一致成立的问题，最终得出Marcinkiewicz-Zygmund SLLN的分布一致推广。 这些结果可以看作是Chung的分布一致SLLN的扩展，适用于具有均匀可积的$q^\text{th}$绝对中心矩的随机变量，对于$0 < q < 2$。 此外，我们证明了$q^\text{th}$矩的均匀可积性是SLLN在Marcinkiewicz-Zygmund速率$n^{1/q - 1}$下一致成立的充分必要条件。 这些证明主要依赖于一些熟悉的几乎处处收敛结果的新型分布一致类比，包括Khintchine-Kolmogorov收敛定理、Kolmogorov的三重级数定理、Kronecker引理的随机推广以及Borel-Cantelli引理。 我们还考虑了非同分布的情况。 显示英文摘要

摘要： 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.