On Divergence-Power Inequalities

TL;DR

This paper introduces new divergence-power inequalities for stationary random processes, linking divergence rates, spectral densities, and Gaussian processes, and provides a novel proof of the Shannon EPI using CMMSE relationships.

Contribution

It presents new divergence-power inequalities for stationary processes and offers a novel proof of the Shannon entropy-power inequality based on CMMSE analysis.

Findings

Derived divergence-power inequalities for discrete and continuous processes.

Connected divergence rates with spectral densities and Gaussian processes.

Provided a new proof of Shannon's EPI using CMMSE in Gaussian channels.

Abstract

Expressions for (EPI Shannon type) Divergence-Power Inequalities (DPI) in two cases (time-discrete and band-limited time-continuous) of stationary random processes are given. The new expressions connect the divergence rate of the sum of independent processes, the individual divergence rate of each process, and their power spectral densities. All divergences are between a process and a Gaussian process with same second order statistics, and are assumed to be finite. A new proof of the Shannon entropy-power inequality EPI, based on the relationship between divergence and causal minimum mean-square error (CMMSE) in Gaussian channels with large signal-to-noise ratio, is also shown.