Entropy rate of product of independent processes

Abstract

We study the multiplicative version of the classical Furstenberg's filtering problem, where instead of the sum $\mathbf{X}+\mathbf{Y}$ one considers the product $\mathbf{X}\cdot \mathbf{Y}$ ($\mathbf{X}$ and $\mathbf{Y}$ are bilateral, real, finitely-valued, stationary independent processes, $\mathbf{Y}$ is taking values in $\{0,1\}$). We provide formulas for $\mathbf{H}(\mathbf{X}\cdot \mathbf{Y}|\mathbf{Y})$. As a consequence, we show that if $\mathbf{H}(\mathbf{X})>\mathbf{H}(\mathbf{Y})=0$ and $\mathbf{X}⨿\mathbf{Y}$, then $\mathbf{H}(\mathbf{X}\cdot \mathbf{Y})<\mathbf{H}(\mathbf{X})$ (and thus $\mathbf{X}$ cannot be filtered out from $\mathbf{X}\cdot \mathbf{Y}$) whenever $\mathbf{X}$ is not bilaterally deterministic, $\mathbf{Y}$ is ergodic and $\mathbf{Y}$ first return to $1$ can take arbitrarily long with positive probability. On the other hand, if almost surely $\mathbf{Y}$ visits $1$ along an infinite arithmetic progression of a fixed difference (with possibly some more visits in between) then we can find $\mathbf{X}$ that is not bilaterally deterministic and such that $\mathbf{H}(\mathbf{X}\cdot \mathbf{Y})=\mathbf{H}(\mathbf{X})$. As a consequence, a $\mathcal{B}$-free system $(X_{\eta },S)$ is proximal if and only if there is always an entropy drop $h(\kappa \ast {\nu }_{\eta })<h(\kappa )$ for any $\kappa$ corresponding to a non-bilaterally deterministic process of positive entropy. These results partly settle some open problems on invariant measures for $\mathcal{B}$-free systems.

arXiv comment25 pages. The paper has been restructured. Main changes: relevant comments on other results on filtering a signal were added, the main technical formula is now stated (and proved) in a stronger form than before, which results in more applications, in particular, new results on B-free systems were added

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