Quantitative Generalized Central Limit Theorems with Self-Decomposable and Symmetric Stable Limiting Laws

Khái niệm cốt lõi

This work provides new stability estimates for centered non-degenerate self-decomposable probability measures on Rd with finite second moment and for non-degenerate symmetric α-stable probability measures on Rd with α ∈ (1, 2). The proofs rely on Stein's method, closed forms techniques, and weighted Poincaré-type inequalities. The results yield explicit rates of convergence in Wasserstein-type distances for several instances of generalized central limit theorems.

Tóm tắt

The content presents new stability estimates for two classes of probability measures on Rd: centered non-degenerate self-decomposable measures with finite second moment, and non-degenerate symmetric α-stable measures with α ∈ (1, 2).
For the self-decomposable case:

The stability estimate is obtained using Stein's method, closed forms techniques, and a new weighted Poincaré-type inequality.
The estimate provides an upper bound on the Wasserstein-2 distance between the target self-decomposable measure and a given probability measure, in terms of the moments of their difference and the Stein kernel.
For the symmetric α-stable case with α ∈ (1, 2):

The stability estimate is derived using the generator approach to Stein's method and a smooth truncation argument.
The estimate bounds the 1-Wasserstein distance between the target symmetric α-stable measure and a given probability measure, in terms of certain integrals involving the weight function and the α-stable Lévy measure.
As applications, the work derives sharp quantitative rates of convergence in Wasserstein-type distances for several instances of generalized central limit theorems with self-decomposable and symmetric stable limiting laws. In particular, a n^(1-2/α) rate is obtained in 1-Wasserstein distance when the target law is a non-degenerate symmetric α-stable one with α ∈ (1, 2).
The non-degenerate symmetric Cauchy (α = 1) case is also studied in depth from a spectral perspective, leading to a n^(-1) rate of convergence when the initial law is a certain instance of layered stable distributions.

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Thông tin chi tiết chính được chắt lọc từ

Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods

by Benjamin Arr... lúc arxiv.org 10-01-2024

https://arxiv.org/pdf/2305.14995.pdf

Yêu cầu sâu hơn

How can the stability estimates and quantitative convergence rates be extended to other classes of limiting laws beyond self-decomposable and symmetric stable distributions?

The stability estimates and quantitative convergence rates established for self-decomposable and symmetric stable distributions can potentially be extended to other classes of limiting laws by leveraging the underlying structural properties of these distributions. One promising approach is to identify additional classes of infinitely divisible distributions that share similar characteristics with self-decomposable laws, such as the existence of a well-defined Lévy measure and the ability to express their characteristic functions in a suitable form.
For instance, one could explore the extension of these results to classes of distributions characterized by specific tail behaviors or moment conditions, such as subexponential or regularly varying distributions. By employing techniques from Stein's method, one can derive stability estimates for these distributions by constructing appropriate Stein kernels that capture the essence of their convergence properties.
Moreover, the use of weighted Poincaré-type inequalities, as introduced in the context of self-decomposable laws, could be adapted to other classes of distributions that exhibit similar functional inequalities. This would involve establishing the necessary conditions for the existence of such inequalities in the new context, thereby allowing for the computation of convergence rates in Wasserstein distances.

Can the techniques developed in this work be adapted to obtain quantitative results in infinite-dimensional settings, such as for generalized central limit theorems in Hilbert spaces?

Yes, the techniques developed in this work can be adapted to obtain quantitative results in infinite-dimensional settings, particularly in the context of generalized central limit theorems in Hilbert spaces. The foundational principles of Stein's method, which focus on assessing the distance between probability measures, can be extended to infinite-dimensional spaces by utilizing the framework of functional analysis and the properties of Hilbert spaces.
In this adaptation, one would need to consider the appropriate definitions of convergence and distance metrics, such as the Wasserstein distance in infinite dimensions. The construction of Stein kernels would also require careful consideration of the structure of the Hilbert space, ensuring that the kernels are well-defined and satisfy the necessary properties for the application of Stein's method.
Additionally, the spectral analysis techniques employed for non-local operators in finite dimensions can be translated to the infinite-dimensional setting, allowing for the exploration of the spectral properties of operators associated with the limiting distributions in Hilbert spaces. This could lead to new insights into the convergence behavior of sequences of random variables in these spaces, thereby enriching the theory of generalized central limit theorems in infinite dimensions.

What are the potential applications of the spectral analysis of the non-local operators associated with non-degenerate symmetric Cauchy measures, beyond the stability estimates considered here?

The spectral analysis of non-local operators associated with non-degenerate symmetric Cauchy measures has several potential applications that extend beyond the stability estimates discussed in the context of this work.

Ergodic Theory: The spectral properties of these operators can provide insights into the long-term behavior of stochastic processes driven by Cauchy measures. This is particularly relevant in ergodic theory, where understanding the spectral gap and the associated semigroups can inform the mixing properties of the underlying processes.

Partial Differential Equations (PDEs): The non-local operators related to Cauchy measures can be linked to certain types of non-local PDEs. The spectral analysis can help in characterizing the solutions to these equations, particularly in understanding the regularity and existence of solutions under various boundary conditions.

Statistical Mechanics: In the context of statistical mechanics, the spectral properties of non-local operators can be used to study phase transitions and critical phenomena. The behavior of these operators can reveal information about the stability of certain states and the nature of phase transitions in systems modeled by Cauchy distributions.

Financial Mathematics: The Cauchy distribution is often used to model heavy-tailed phenomena in finance. The spectral analysis of associated non-local operators can aid in the development of pricing models for financial derivatives, particularly those sensitive to extreme events.

Machine Learning: In the realm of machine learning, understanding the spectral properties of non-local operators can enhance the design of algorithms that rely on probabilistic models with heavy-tailed distributions, improving their robustness and performance in the presence of outliers.

Overall, the spectral analysis of non-local operators associated with non-degenerate symmetric Cauchy measures opens up a wide array of applications across various fields, highlighting the interdisciplinary nature of the results obtained in this work.