Product gales and Finite state dimension

TL;DR

This paper introduces product gales and multi-bet finite-state dimension, providing new characterizations of Hausdorff dimension and finite-state dimension through entropy rates and automata-based methods.

Contribution

It defines product gales and multi-bet finite-state dimension, showing their equivalence to existing notions and offering new characterizations via entropy rates and automata theory.

Findings

Product gales characterize Hausdorff dimension.

Multi-bet finite-state dimension equals finite-state dimension.

Automata-based proof of entropy rate equivalence.

Abstract

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.