Convergence rate in precise asymptotics for Davis law of large numbers
L Kong, H Dai - Statistics & Probability Letters, 2016 - Elsevier
Let {X, X n, n≥ 1} be a sequence of iid random variables with E [X]= 0 and E [X 2]= σ
2∈(0,∞), and set S n=∑ k= 1 n X k, n≥ 1. For any δ≥ 0, let γ δ= lim n→∞(∑ j= 1 n (log j) δ
j−(log n) δ+ 1 δ+ 1) and η δ=∑ n= 1∞(log n) δ n P (S n= 0). Under the moment condition E
[X 2 (log (1+∣ X∣)) 1+ δ]<∞, we prove that lim ϵ↘ 0 [∑ n= 1∞(log n) δ n P (∣ S n∣≥ ϵ n
log n)− E [∣ N∣ 2 δ+ 2] δ+ 1 σ 2 δ+ 2 ϵ−(2 δ+ 2)]= γ δ− η δ, which refines Theorem 3 of Gut
and Spătaru (2000a).
2∈(0,∞), and set S n=∑ k= 1 n X k, n≥ 1. For any δ≥ 0, let γ δ= lim n→∞(∑ j= 1 n (log j) δ
j−(log n) δ+ 1 δ+ 1) and η δ=∑ n= 1∞(log n) δ n P (S n= 0). Under the moment condition E
[X 2 (log (1+∣ X∣)) 1+ δ]<∞, we prove that lim ϵ↘ 0 [∑ n= 1∞(log n) δ n P (∣ S n∣≥ ϵ n
log n)− E [∣ N∣ 2 δ+ 2] δ+ 1 σ 2 δ+ 2 ϵ−(2 δ+ 2)]= γ δ− η δ, which refines Theorem 3 of Gut
and Spătaru (2000a).
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