Some remarks on the rate of convergence to stable laws

Juozulynas, Algirdas; Paulauskas, Vygantas

doi:10.1007/bf02465818

Cited by 20 publications

(10 citation statements)

References 10 publications

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“…The convergence rate in Theorem 1.4 matches the optimal one in Kolmogorov distance found by Hall [22], see more details in Remark 1.5. When X 1 is out of the scope of normal attraction of stable law, we also found a convergence rate, which is in the same order as the best rate reported in [24].…”

Section: Motivation and Main Resultssupporting

confidence: 81%

Approximation to stable law by the Lindeberg principle

Chen

2019

Journal of Mathematical Analysis and Applications

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Section: Motivation and Main Resultssupporting

confidence: 81%

Approximation to stable law by the Lindeberg principle

Chen

2019

Journal of Mathematical Analysis and Applications

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“…In Appendix B, we also consider an example where the common distribution has the regularly varying density p(x) = α 2 e α 2(1+α) log |x| |x| α+1 1 [e,∞) (|x|), which is not in the domain of normal attraction of a stable law. We obtain the same convergence rate (log n) −1 as [28] for the case p(x) =…”

Section: Examples and Applicationsupporting

confidence: 55%

“…The discussion around rates of convergence in the generalized CLT has been made by many authors. Using the distance d Kol , Hall [26] gave two-sided bounds (tight under some assumption), see also [28] where the same distance was used. In [30,16] were discussed the L ∞ or L 1 rate of convergence of the density function of the approximating sequence.…”

Section: Introductionmentioning

confidence: 99%

Non-integrable stable approximation by Stein's method

Chen

Nourdin

et al. 2019

Preprint

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We develop Stein's method for α-stable approximation with α ∈ (0, 1], continuing the recent line of research by Xu [40] and Chen, Nourdin and Xu [11] in the case α ∈ (1, 2). The main results include an intrinsic upper bound for the error of the approximation in a variant of Wasserstein distance that involves the characterizing differential operators for stable distributions, and an application to the generalized central limit theorem. Due to the lack of first moment for the approximating sequence in the latter result, we appeal to an additional truncation procedure and investigate fine regularity properties of the solution to Stein's equation.

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“…To do this, first we have to deal with the one dimensional case. Similar calculations were done previously, see, for example [12] and [14]. However, in these articles only one dimensional, non-lattice distributions were considered.…”

Section: Local Limit Theorem With Remainder Termmentioning

confidence: 72%

Recurrence properties of a special type of Heavy-Tailed Random Walk

Nándori

2016

Preprint

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In the proof of the invariance principle for locally perturbed periodic Lorentz process with finite horizon, a lot of delicate results were needed concerning the recurrence properties of its unperturbed version. These were analogous to the similar properties of Simple Symmetric Random Walk. However, in the case of Lorentz process with infinite horizon, the analogous results for the corresponding random walk are not known, either. In this paper, these properties are ascertained for the appropriate random walk (this happens to be in the non normal domain of attraction of the normal law). As a tool, an estimation of the remainder term in the local limit theorem for the corresponding random walk is computed.Recently, D. Paulin and D. Szász proved ([16]) that the random walk, which is very similar to the

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Some remarks on the rate of convergence to stable laws

Cited by 20 publications

References 10 publications

Approximation to stable law by the Lindeberg principle

Approximation to stable law by the Lindeberg principle

Non-integrable stable approximation by Stein's method

Recurrence properties of a special type of Heavy-Tailed Random Walk

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