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| Format: | Preprint |
| Veröffentlicht: |
2025
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| Online-Zugang: | https://arxiv.org/abs/2504.02681 |
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| _version_ | 1866908299497046016 |
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| author | Anshelevich, Michael Nguyen, Anh |
| author_facet | Anshelevich, Michael Nguyen, Anh |
| contents | Let $\{A_{i,n}\}$ be a triangular array of elements in a Banach algebra, whose norms do not grow too fast, and whose row averages converge to $A$. Let $σ\in S(n)$ be a permutation drawn uniformly at random. If the array only contains $o(n / \log n)$ distinct elements, then almost surely, for all $0 < s < t < 1$, the permuted product of their exponentials $\prod_{i = [s n]}^{[t n]} e^{A_{σ(i),n}/n}$ converges in norm to $e^{(t - s) A}$. For an array of finite-dimensional matrices, convergence holds without this restriction. The proof of the latter result consists of an estimate valid in a general Banach algebra, and an application of a matrix concentration inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Convergence of permuted products of exponentials Anshelevich, Michael Nguyen, Anh Functional Analysis Probability 15A16 Let $\{A_{i,n}\}$ be a triangular array of elements in a Banach algebra, whose norms do not grow too fast, and whose row averages converge to $A$. Let $σ\in S(n)$ be a permutation drawn uniformly at random. If the array only contains $o(n / \log n)$ distinct elements, then almost surely, for all $0 < s < t < 1$, the permuted product of their exponentials $\prod_{i = [s n]}^{[t n]} e^{A_{σ(i),n}/n}$ converges in norm to $e^{(t - s) A}$. For an array of finite-dimensional matrices, convergence holds without this restriction. The proof of the latter result consists of an estimate valid in a general Banach algebra, and an application of a matrix concentration inequality. |
| title | Convergence of permuted products of exponentials |
| topic | Functional Analysis Probability 15A16 |
| url | https://arxiv.org/abs/2504.02681 |