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Recent studies found that many channels are affected by additive noise thatis impulsive in nature and is best explained by heavy-tailed symmetricalpha-stable distributions. Dealing with impulsive noise environments comeswith an added complexity with respect to the standard Gaussian environment: thealpha-stable probability density functions have an infinite second moment andthe "nice" Hilbert space structure of the space of random variables having afinite second moment is lost along with its tools and methodologies. This isindeed the case in estimation theory where classical tools to quantifyperformance of an estimator are tightly related to the assumption of finitevariance variables. In alpha-stable environments, expressions such as the meansquare error and the Cramer-Rao bound are hence problematic. In this work, wetackle the parameter estimation problem in impulsive noise environments anddevelop novel tools that are tailored to the alpha-stable and heavy-tailednoise environments, tools that coincide with the standard ones adopted in theGaussian setup, namely a generalized "power" measure and a generalized Fisherinformation. We generalize known information inequalities commonly used in theGaussian context: the de Bruijn's identity, the data processing inequality, theFisher information inequality, the isoperimetric inequality for entropies andthe Cramer-Rao bound. Additionally, we derive upper bounds on the differentialentropy of independent sums having a stable component. Finally, the new "power"measure is used to shed some light on the additive alpha-stable noise channelcapacity in a setup that generalizes the linear average power constrained AWGNchannel. Our theoretical findings are paralleled with numerical evaluations ofvarious quantities and bounds using developed {\em Matlab} packages.

Information Measures, Inequalities and Performance Bounds for Parameter Estimation in Impulsive Noise Environments