Convergence rate in precise asymptotics for the law of the iterated logarithm

Kong, Lingtao

doi:10.1007/s10986-016-9321-4

Cited by 3 publications

(2 citation statements)

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“…In this paper, using the rate of convergence to the normal distribution and Fubini theorem, under some suitable conditions, the convergence rates in precise asymptotics for the complete convergence have been discussed with more general boundary functions. The result extends and generalizes the corresponding results of Gut and Steinebach [1], Kong [2], and Kong and Dai [3]. However, this paper has only studied the convergence rates for complete convergence.…”

Section: Discussionsupporting

confidence: 84%

“…Remark 1.7 Obviously, Corollary 1.3 with s = 1 2(δ+1) extends Theorem 1.1 in Kong and Dai [3] with the scope of δ from δ ≥ 0 to δ > -1; Corollary 1.4 with s = 1 2(b+1) extends Theorem 1 from Kong [2] with the scope of b from b = 0 to b > -1 and the moment condition from E|X| q < ∞ (2 < q ≤ 3) to EX 2 (log(1 + |X|)) < ∞; Corollary 1.5 with s = 2-p 2(r-p) extends Theorem 2.2(a) from Gut and Steinebach [1] with the scope of r, p from 1 ≤ p < 2, p < r < 3p/2 to 0 < p < 2, p < r < 3p/2. Therefore our results extend the known results.…”

Section: Theorem 11 Let G(x) Be a Positive And Twice Differentiable mentioning

confidence: 75%

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A note on the convergence rates in precise asymptotics

Zhang

2019

J Inequal Appl

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Let {X, X n , n ≥ 1} be a sequence of i.i.d. random variables with EX = 0, EX 2 = σ 2. Set S n = n k=1 X k and let N be the standard normal random variable. Let g(x) be a positive and twice differentiable function on [n 0 , ∞) such that g(x) ∞, g (x) 0 as x → ∞. In this short note, under some suitable conditions on both X and g(x), we establish the following convergence rates in precise asymptotics lim ε 0 ∞ n=n 0 g (n)P |S n | σ √ n ≥ εg s (n)-ε-1/s E|N | 1/s = γη, where γ = lim n→∞ (n k=n 0 g (k)-g(n)), η = ∞ n=n 0 g (n)P{S n = 0}. It can describe the relations among the boundary function, weighting function, convergence rate and the limit value in studies of complete convergence. The result extends and generalizes the corresponding results of Gut and Steinebach (Ann. Univ. Sci.

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Section: Discussionsupporting

confidence: 84%