Repetition and recurrence times: Dual statements and summable mixing rates
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916934052741120 |
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| author | Dębowski, Łukasz |
| author_facet | Dębowski, Łukasz |
| contents | By an analogy to the duality between the recurrence time and the longest match length, we introduce a quantity dual to the maximal repetition length, which we call the repetition time. Extending prior results, we sandwich the repetition time in terms of unconditional and conditional min-entropies. The upper bound holds if the mixing rate $ϕ(n)$ is summable, whereas the lower bound only assumes stationarity. Our reasoning makes a repeated use of dualities between so-called times and so-called counts that generalize the duality of the recurrence time and the longest match length. We also discuss the analogy of these results with the Wyner-Ziv/Ornstein-Weiss theorem, which sandwiches the recurrence time in terms of Shannon entropies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_14703 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Repetition and recurrence times: Dual statements and summable mixing rates Dębowski, Łukasz Information Theory Probability 60G10 (Primary) 94A17 (Secondary) By an analogy to the duality between the recurrence time and the longest match length, we introduce a quantity dual to the maximal repetition length, which we call the repetition time. Extending prior results, we sandwich the repetition time in terms of unconditional and conditional min-entropies. The upper bound holds if the mixing rate $ϕ(n)$ is summable, whereas the lower bound only assumes stationarity. Our reasoning makes a repeated use of dualities between so-called times and so-called counts that generalize the duality of the recurrence time and the longest match length. We also discuss the analogy of these results with the Wyner-Ziv/Ornstein-Weiss theorem, which sandwiches the recurrence time in terms of Shannon entropies. |
| title | Repetition and recurrence times: Dual statements and summable mixing rates |
| topic | Information Theory Probability 60G10 (Primary) 94A17 (Secondary) |
| url | https://arxiv.org/abs/2306.14703 |