Entropy rate of product of independent processes

Abstract: The entropy of the product of stationary processes is related to Furstenberg’s filtering problem. In its classical version one deals with the sum $\bm{X}+\bm{Y}$, where $\bm{X}$ corresponds to the signal and $\bm{Y}$ to the noise. In his seminal paper from 1967, Furstenberg showed that under the natural assumption of the disjointness of underlying dynamical systems, the information about $\bm{X}$ can be retrieved from $\bm{X}+\bm{Y}$. Instead of the sum, we study the product $\bm{X}\cdot\bm{Y}$. We give a formula for the entropy rate of $\bm{X}\cdot\bm{Y}$ (relative to that of $\bm{Y}$, for $\bm{X}$ and $\bm{Y}$ being independent). As a consequence, $\bm{X}$ cannot be recovered from $\bm{X}\cdot\bm{Y}$ for a wide class of positive entropy processes, including exchangeable processes, Markov chains and weakly Bernoulli processes. Moreover, we answer some open problems on the dynamics of $\mathscr{B}$-free systems (including the square-free system given by the square of the Moebius function). The talk is based on joint work with Michał Lemańczyk, see arxiv.org/pdf/2004.07648.pdf

dynamical systems

Audience: researchers in the topic

Dynamical systems seminar at the Jagiellonian University