Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866908567605346304 |
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| author | Waudby-Smith, Ian Larsson, Martin Ramdas, Aaditya |
| author_facet | Waudby-Smith, Ian Larsson, Martin Ramdas, Aaditya |
| contents | We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation Waudby-Smith, Ian Larsson, Martin Ramdas, Aaditya Probability Statistics Theory We present nonasymptotic concentration inequalities for sums of independent and identically distributed random variables that yield asymptotic strong Gaussian approximations of Komlós, Major, and Tusnády (KMT) [1975,1976]. The constants appearing in our inequalities are either universal or explicit, and thus as corollaries, they imply distribution-uniform generalizations of the aforementioned KMT approximations. In particular, it is shown that uniform integrability of a random variable's $q^{\text{th}}$ moment is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $o(n^{1/q})$ for $q > 2$ and that having a uniformly lower bounded Sakhanenko parameter -- equivalently, a uniformly upper-bounded Bernstein parameter -- is both necessary and sufficient for the KMT approximations to hold uniformly at the rate of $O(\log n)$. Instantiating these uniform results for a single probability space yields the analogous results of KMT exactly. |
| title | Nonasymptotic and distribution-uniform Komlós-Major-Tusnády approximation |
| topic | Probability Statistics Theory |
| url | https://arxiv.org/abs/2502.06188 |