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Entropy Rate of Product of Independent Processes by Kułaga-Przymus, Joanna; Lemańczyk, Michał is a scholarly article available to read on EtoBox.

What is Entropy Rate of Product of Independent Processes about?

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 $\mathbf{X}\amalg \mathbf{Y}$, 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 $\mathscr{B}$-f

Author

Kułaga-Przymus, Joanna; Lemańczyk, Michał

Published

2020

Language

EN