2024/02/01 by Ian Waudby-Smith, Martin Larsson, Waudby-Smith, Ian +3 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Statistics Theory (math.ST) #Stochastic processes and financial applications
paper · pdf · doi:10.48550/arxiv.2402.00713
openalex publication_date 2024/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 qth absolute central moments for 0 < q < 2. Furthermore, we show that uniform integrability of the qth moment is both sufficient and necessary for the SLLN to hold uniformly at the Marcinkiewicz-Zygmund rate of n1/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.