Moments of finitary factor maps between Bernoulli processes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916938811179008 |
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| author | Gabor, Uri |
| author_facet | Gabor, Uri |
| contents | The problem of what moments can exist for the coding radius of a finitary map between two i.i.d. processes, has been extensively studied in the case of $\mathbb{Z}$-processes. Here we treat this problem for factor maps between $\mathbb{Z}^{d}$-processes ($d>1$). By modeling the homomorphism with a map between spaces of finite sequences, we extend Harvey and Peres' result, showing that for a finitary homomorphism between two i.i.d. processes of equal entropy, if the coding radius of the map has a finite $\frac{d}{2}$-moment, then the two processes share the same informational variance. We use our modeling technique to prove a "Schmidt-type theorem" - that in case the above homomorphism has a coding radius of exponential tails, then the two processes are essentially the same. This result appears to be new even for the one-dimensional case, addressing a question of Angel and Spinka. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_06018 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Moments of finitary factor maps between Bernoulli processes Gabor, Uri Probability Information Theory Dynamical Systems 37A35, 37A50, 60G10 The problem of what moments can exist for the coding radius of a finitary map between two i.i.d. processes, has been extensively studied in the case of $\mathbb{Z}$-processes. Here we treat this problem for factor maps between $\mathbb{Z}^{d}$-processes ($d>1$). By modeling the homomorphism with a map between spaces of finite sequences, we extend Harvey and Peres' result, showing that for a finitary homomorphism between two i.i.d. processes of equal entropy, if the coding radius of the map has a finite $\frac{d}{2}$-moment, then the two processes share the same informational variance. We use our modeling technique to prove a "Schmidt-type theorem" - that in case the above homomorphism has a coding radius of exponential tails, then the two processes are essentially the same. This result appears to be new even for the one-dimensional case, addressing a question of Angel and Spinka. |
| title | Moments of finitary factor maps between Bernoulli processes |
| topic | Probability Information Theory Dynamical Systems 37A35, 37A50, 60G10 |
| url | https://arxiv.org/abs/2509.06018 |