A functional Hungarian construction for sums of independent random variables
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866912161947713536 |
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| author | Grama, Ion Nussbaum, Michael |
| author_facet | Grama, Ion Nussbaum, Michael |
| contents | We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is taken outside the probability. This form is a prerequisite for the Komlós-Major-Tusnády inequality in the space of bounded functionals $l^{\infty }(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1/2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the Hölder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_15043 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A functional Hungarian construction for sums of independent random variables Grama, Ion Nussbaum, Michael Probability 60F17, 60F99, 62G07 We develop a Hungarian construction for the partial sum process of independent non-identically distributed random variables. The process is indexed by functions $f$ from a class $\mathcal{H}$, but the supremum over $f\in $ $\mathcal{H}$ is taken outside the probability. This form is a prerequisite for the Komlós-Major-Tusnády inequality in the space of bounded functionals $l^{\infty }(\mathcal{H})$, but contrary to the latter it essentially preserves the classical $n^{-1/2}\log n$ approximation rate over large functional classes $\mathcal{H}$ such as the Hölder ball of smoothness $1/2$. This specific form of a strong approximation is useful for proving asymptotic equivalence of statistical experiments. |
| title | A functional Hungarian construction for sums of independent random variables |
| topic | Probability 60F17, 60F99, 62G07 |
| url | https://arxiv.org/abs/2412.15043 |