Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913500236873728 |
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| author | Butzek, Marius Eichelsbacher, Peter |
| author_facet | Butzek, Marius Eichelsbacher, Peter |
| contents | In this paper we obtain non-uniform Berry-Esseen bounds for normal approximations by the Malliavin-Stein method. The techniques rely on a detailed analysis of the solutions of Stein's equations and will be applied to functionals of a Gaussian process like multiple Wiener-Itô integrals, to Poisson functionals as well as to the Rademacher chaos expansion. Second-order Poincaré inequalities for normal approximation of these functionals are connected with non-uniform bounds as well. As applications, elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process, like the discretized version of the quadratic variation of a fractional Brownian motion, are considered. Moreover we consider subgraph counts in random geometric graphs as an example of Poisson $U$-statistics, as well as subgraph counts in the Erdős-Rényi random graph and infinite weighted 2-runs as examples of functionals of Rademacher variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_09439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes Butzek, Marius Eichelsbacher, Peter Probability In this paper we obtain non-uniform Berry-Esseen bounds for normal approximations by the Malliavin-Stein method. The techniques rely on a detailed analysis of the solutions of Stein's equations and will be applied to functionals of a Gaussian process like multiple Wiener-Itô integrals, to Poisson functionals as well as to the Rademacher chaos expansion. Second-order Poincaré inequalities for normal approximation of these functionals are connected with non-uniform bounds as well. As applications, elements living inside a fixed Wiener chaos associated with an isonormal Gaussian process, like the discretized version of the quadratic variation of a fractional Brownian motion, are considered. Moreover we consider subgraph counts in random geometric graphs as an example of Poisson $U$-statistics, as well as subgraph counts in the Erdős-Rényi random graph and infinite weighted 2-runs as examples of functionals of Rademacher variables. |
| title | Non-uniform Berry--Esseen bounds for Gaussian, Poisson and Rademacher processes |
| topic | Probability |
| url | https://arxiv.org/abs/2409.09439 |