Mutual Dimension and Random Sequences

TL;DR

This paper explores the relationship between mutual dimensions of infinite sequences and coupled randomness, providing explicit formulas, conditions for independence, and generalizations of mutual dimensions within algorithmic information theory.

Contribution

It introduces explicit formulas for mutual dimensions of coupled random sequences and establishes conditions for their independence and meaningful generalizations.

Findings

Explicit formulas for mutual dimensions under certain measures

Condition that zero mutual dimension does not imply independence

Generalizations of mutual dimensions align with existing theoretical frameworks

Abstract

If $S$ and $T$ are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions $mdim(S:T)$ and $Mdim(S:T)$ are the upper and lower densities of the algorithmic information that is shared by $S$ and $T$. In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences $R_1$ and $R_2$ with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions $mdim(R_1:R_2)$ and $Mdim(R_1:R_2)$, and we show that the condition $Mdim(R_1:R_2) = 0$ is necessary but not sufficient for $R_1$ and $R_2$ to be independently random. We also identify conditions under which Billingsley generalizations of the mutual dimensions $mdim(S:T)$ and $Mdim(S:T)$ can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the "correct" relationships with the Billingsley generalizations of $dim(S)$, $Dim(S)$, $dim(T)$, and $Dim(T)$ that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.