Product gales and Finite state dimension
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910820164698112 |
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| author | S, Akhil |
| author_facet | S, Akhil |
| contents | In this work, we introduce the notion of product gales, which is the modification of an $s$-gale such that $k$ separate bets can be placed at each symbol. The product of the bets placed are taken into the capital function of the product-gale. We show that Hausdorff dimension can be characterised using product gales.
A $k$-bet finite-state gambler is one that can place $k$ separate bets at each symbol. We call the notion of finite-state dimension, characterized by product gales induced by $k$-bet finite-state gamblers, as multi-bet finite-state dimension. Bourke, Hitchcock and Vinodchandran gave an equivalent characterisation of finite state dimension by disjoint block entropy rates. We show that multi-bet finite state dimension can be characterised using sliding block entropy rates. Further, we show that multi-bet finite state dimension can also be charatcterised by disjoint block entropy rates.
Hence we show that finite state dimension and multi-bet finite state dimension are the same notions, thereby giving a new characterisation of finite state dimension using $k$-bet finite state $s$-gales. We also provide a proof of equivalence between sliding and disjoint block entropy rates, providing an alternate, automata based proof of the result by Kozachinskiy, and Shen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_06208 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Product gales and Finite state dimension S, Akhil Information Theory 68P30 68Q45 In this work, we introduce the notion of product gales, which is the modification of an $s$-gale such that $k$ separate bets can be placed at each symbol. The product of the bets placed are taken into the capital function of the product-gale. We show that Hausdorff dimension can be characterised using product gales. A $k$-bet finite-state gambler is one that can place $k$ separate bets at each symbol. We call the notion of finite-state dimension, characterized by product gales induced by $k$-bet finite-state gamblers, as multi-bet finite-state dimension. Bourke, Hitchcock and Vinodchandran gave an equivalent characterisation of finite state dimension by disjoint block entropy rates. We show that multi-bet finite state dimension can be characterised using sliding block entropy rates. Further, we show that multi-bet finite state dimension can also be charatcterised by disjoint block entropy rates. Hence we show that finite state dimension and multi-bet finite state dimension are the same notions, thereby giving a new characterisation of finite state dimension using $k$-bet finite state $s$-gales. We also provide a proof of equivalence between sliding and disjoint block entropy rates, providing an alternate, automata based proof of the result by Kozachinskiy, and Shen. |
| title | Product gales and Finite state dimension |
| topic | Information Theory 68P30 68Q45 |
| url | https://arxiv.org/abs/2502.06208 |