  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.
