A kernel bound for non-symmetric stable distribution and its applications

Jin, Xinghu; Li, Xiang; Lu, Jianya

doi:10.1016/j.jmaa.2020.124063

Cited by 6 publications

(11 citation statements)

References 13 publications

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“…Further, Jin et al [21] and Chen et al [11] extend Xu's idea [39], and develop Stein's method for asymmetric α-stable distributions with α ∈ (1, 2). In [21], the authors obtain a kernel discrepancy type bound as (2.10), and derive the convergence rate n − 2−α α for asymmetric α-stable approximations in the Wasserstein-1 distance.…”

Section: Literature Reviewmentioning

confidence: 99%

“…In this direction, Xu [39] developed Stein's method for symmetric α-stable distributions with α ∈ (1, 2). Chen et al [11] and Jin et al [21] extended Xu's idea [39] and developed Stein's method for asymmetric α-stable distributions with α ∈ (1, 2). Later, Chen et al [12] developed Stein's method for multivariate α-stable distributions with α ∈ (1, 2).…”

Section: Introductionmentioning

confidence: 99%

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A unified approach to Stein’s method for stable distributions

Upadhye

Barman²

2022

Probab. Surveys

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In this article, we first review the connection between Lévy processes and infinitely divisible random variables, and discuss the classification of infinitely divisible distributions. Next, we establish a Stein identity for an infinitely divisible random variable via the Lévy-Khintchine representation of the characteristic function. In particular, we establish and unify the Stein identities for an α-stable random variable available in the existing literature. Next, we derive the solutions of the Stein equations and its regularity estimates. Further, we derive error bounds for α-stable approximations, and also obtain rates of convergence results in Wasserstein-δ, δ < α for α ∈ (0, 1) and Wasserstein-type distances for α ∈ (1, 2). Finally, we compare these results with existing literature.

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Section: Literature Reviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A unified approach to Stein’s method for stable distributions

Upadhye

Barman²

2022

Probab. Surveys

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“…The deviation performance of this estimator is much better than X. Catoni's idea has been broadly applied to many research problems, see for instance [1,15,5,6,7,11,12,17]. The finite variance assumption plays an important role in Catoni's analysis, but it rules out many interesting distributions such as Pareto law [10,16,4,8], which describes the distributions of wealth and social networks. We generalize Catoni's M-estimator to the case in which samples can have finite α-th moment with α ∈ (1, 2).…”

Section: Introductionmentioning

confidence: 99%

A generalized Catoni’s M-estimator under finite α-th moment assumption with α∈(1,2)

Chen

Jin

et al. 2021

Electron. J. Statist.

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We generalize Catoni's M-estimator, put forward in [3] by Catoni under finite variance assumption, to the case in which distributions can have finite α-th moment with α ∈ (1, 2). Our approach, inspired by the Taylor-like expansion developed in [4], is via slightly modifying the influence function ϕ in [3]. A deviation bound is established for this generalized estimator, and coincides with that in [3] as α ↑ 2. Experiment shows that our M-estimator performs better than the empirical mean, the smaller the α is, the better the performance will be. As an application, we study an 1 regression considered by Zhang et al. [19], who assumed that samples have finite variance, under finite α-th moment assumption with α ∈ (1, 2).

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“…On the other hand, if Θ is a singleton, (X t ) t≥0 is a classical Lévy process with triplet Θ, and X 1 is an α-stable random variable. The convergence rate of the classical α-stable central limit theorem has been studied in the Kolmogorov distance (see, e.g., [13, 15-17, 21, 24]) and in the Wasserstein-1 distance or the smooth Wasserstein distance (see, e.g., [1,10,11,23,30,38]). The first type is proved by the characteristic functions that do not exist in the sublinear framework, while the second type relies on Stein's method, which fails under the sublinear setting.…”

Section: Introductionmentioning

confidence: 99%

A monotone scheme for nonlinear partial integro-differential equations with the convergence rate of $α$-stable limit theorem under sublinear expectation

Hu,

Jiang,

Liang

2021

Preprint

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In this paper, we propose a monotone approximation scheme for a class of fully nonlinear partial integro-differential equations (PIDEs) which characterize the nonlinear α-stable Lévy processes under sublinear expectation space with α ∈ (1, 2). Two main results are obtained: (i) the error bounds for the monotone approximation scheme of nonlinear PIDEs, and (ii) the convergence rates of a generalized central limit theorem of Bayraktar-Munk for α-stable random variables under sublinear expectation. Our proofs use and extend techniques introduced by Krylov and Barles-Jakobsen.

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A kernel bound for non-symmetric stable distribution and its applications

Cited by 6 publications

References 13 publications

A unified approach to Stein’s method for stable distributions

A unified approach to Stein’s method for stable distributions

A generalized Catoni’s M-estimator under finite α-th moment assumption with α∈(1,2)

A monotone scheme for nonlinear partial integro-differential equations with the convergence rate of $α$-stable limit theorem under sublinear expectation

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