Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914959197208576 |
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| author | Arras, Benjamin |
| author_facet | Arras, Benjamin |
| contents | In these notes, we obtain new stability estimates for centered non-degenerate selfdecomposable probability measures on $\mathbb{R}^d$ with finite second moment and for non-degenerate symmetric $α$-stable probability measures on $\mathbb{R}^d$ with $α\in [1,2)$. These new results are refinements of the corresponding ones available in the literature. The proofs are based on Stein's method for self-decomposable laws, recently developed in a series of papers, and on closed forms techniques together with a new ingredient: weighted Poincaré-type inequalities. As applications, rates of convergence in Wasserstein-type distances are computed for several instances of the generalized central limit theorems (CLTs). 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 $α\in (1,2)$. Finally, the non-degenerate symmetric Cauchy case is studied at length from a spectral point of view. At last, in this Cauchy situation, a $n^{-1}$-rate of convergence is obtained when the initial law is a certain instance of layered stable distributions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_14995 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods Arras, Benjamin Probability Functional Analysis 60E07, 60F05, 31C25, 26D10 In these notes, we obtain new stability estimates for centered non-degenerate selfdecomposable probability measures on $\mathbb{R}^d$ with finite second moment and for non-degenerate symmetric $α$-stable probability measures on $\mathbb{R}^d$ with $α\in [1,2)$. These new results are refinements of the corresponding ones available in the literature. The proofs are based on Stein's method for self-decomposable laws, recently developed in a series of papers, and on closed forms techniques together with a new ingredient: weighted Poincaré-type inequalities. As applications, rates of convergence in Wasserstein-type distances are computed for several instances of the generalized central limit theorems (CLTs). 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 $α\in (1,2)$. Finally, the non-degenerate symmetric Cauchy case is studied at length from a spectral point of view. At last, in this Cauchy situation, a $n^{-1}$-rate of convergence is obtained when the initial law is a certain instance of layered stable distributions. |
| title | Some Notes on Quantitative Generalized CLTs with Self-Decomposable Limiting Laws by Spectral Methods |
| topic | Probability Functional Analysis 60E07, 60F05, 31C25, 26D10 |
| url | https://arxiv.org/abs/2305.14995 |