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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.17794 |
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| _version_ | 1866908723250724864 |
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| author | Iyer, Gautam Venkatraman, Raghavendra |
| author_facet | Iyer, Gautam Venkatraman, Raghavendra |
| contents | Given $N$ i.i.d. samples from a probability measure $μ$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $μ_N \to μ$ in the negative Sobolev space $W^{-α, p}$. When $W^{-α, p}$ contains point measures (i.e. when $αp > (p-1)d$), we show $\mathbf{E} \| μ_N - μ\|_{W^{-α, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < αp \leq d(p-1)$, we prove a similar result for Gaussian regularizations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_17794 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of Empirical Measures for i.i.d. samples in $W^{-α, p}$ Iyer, Gautam Venkatraman, Raghavendra Probability 60B10 (Primary) 60G50, 46E35 (Secondary) Given $N$ i.i.d. samples from a probability measure $μ$ on $\mathbf{R}^d$, we study the rate of convergence of the empirical measure $μ_N \to μ$ in the negative Sobolev space $W^{-α, p}$. When $W^{-α, p}$ contains point measures (i.e. when $αp > (p-1)d$), we show $\mathbf{E} \| μ_N - μ\|_{W^{-α, p}}^p \leq C_d / N^{p/2}$ for an explicit dimensional constant $C_d$, and obtain a Gaussian tail bound. When $0 < αp \leq d(p-1)$, we prove a similar result for Gaussian regularizations. |
| title | Convergence of Empirical Measures for i.i.d. samples in $W^{-α, p}$ |
| topic | Probability 60B10 (Primary) 60G50, 46E35 (Secondary) |
| url | https://arxiv.org/abs/2512.17794 |