2020/07/10 by Deli Li, Yu Miao, Li, Deli +1
Decision Sciences · Mathematics · #Combinatorics #FOS: Mathematics #Law of large numbers #Mathematics #Physics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Random variable #Sequence (biology) #Statistics #Stochastic processes and statistical mechanics #Symmetrization #math.PR
paper · pdf · doi:10.48550/arxiv.2007.05150
published in arXiv (Cornell University) (Cornell University) · 19 pages
arxiv created 2020/07/10 · openalex publication_date 2020/07/10 · arxiv updated 2020/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
Let 0 < p < 2. Let \X, Xn; n ≥ 1\ be a sequence of independent and identically distributed B-valued random variables and set Sn = ∑i=1nXi,~n ≥ 1. In this paper, a supplement to the classical laws of large numbers and the classical large deviations is provided. We show that if Sn/n1/p →ℙ 0, then, for all s > 0, \limsupn → ∞ (1)/(log n) log ℙ(‖Sn ‖ gt; s n1/p ) = - (β - p)/p and \liminfn → ∞ (1)/(log n) log ℙ(‖Sn ‖ gt; s n1/p ) = -(\underlineβ - p)/p, where β = - \limsupt → ∞ (log ℙ(log ‖X‖ gt; t))/(t) ~~and~~\underlineβ = - \liminft → ∞ (log ℙ(log ‖X‖ gt; t))/(t). The main tools employed in proving this result are the symmetrization technique and three powerful inequalities established by Hoffmann-Jørgensen (1974), de Acosta (1981), and Ledoux and Talagrand (1991), respectively. As a special case of this result, the main results of Hu and Nyrhinen (2004) are not only improved, but also extended.