A supplement to the laws of large numbers and the large deviations

2020/07/10 by Deli Li, Yu Miao, Li, Deli +1
Decision Sciences · Mathematics · #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2007.05150

openalex publication_date 2020/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let 0 < p < 2. Let \X, X_n; n ≥ 1\ be a sequence of independent and identically distributed B-valued random variables and set S_n = ∑_i=1ⁿX_i,~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 S_n/n^1/p →_ℙ 0, then, for all s > 0, \limsup_{n → ∞} (1)/(log n) log ℙ(‖S_n ‖ gt; s n^1/p ) = - (β - p)/p and \liminf_{n → ∞} (1)/(log n) log ℙ(‖S_n ‖ gt; s n^1/p ) = -(\underlineβ - p)/p, where β = - \limsup_{t → ∞} (log ℙ(log ‖X‖ gt; t))/(t) ~~and~~\underlineβ = - \liminf_{t → ∞} (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.

Citations

• A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the sum of Observations

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