An information theoretic treatise on univariate alpha-stable distributions -

Abstract

Many communication channels are reasonably modeled to be impaired by additive noise. A Central Limit Theorem (CLT) argument is widely adopted to model the noise as a Gaussian variable. A deeper investigation shows that the CLT motivation leads to noise models that are in general stable and not necessarily Gaussian. This is validated by recent studies suggesting that many channels are affected by additive noise that is impulsive in nature and is best explained by the heavy tailed non-totally skewed alpha-stable distributions. Considering impulsive noise environments comes with an added complexity with respect to the standard Gaussian environment: the alpha-stable probability density functions do not possess closed-form expressions except in some special cases. Furthermore, they have an infinite second moment and the “nice” Hilbert space structure defined by the space of random variables having a finite second moment –which represents the universe in which the Gaussian theory is applicable, is lost along with its tools and methodologies. We study these probability models, their detrimental effect as noise variables and we investigate various bounds on the performance limits in classical problems arising from noisy observations of some quantity of interest. Our approach is from an information theory point of view and some related disciplines: i.We study the channel capacity of channels affected by non-totally skewed alpha-stable noise models and other types of impulsive noise channels. We characterize capacity achieving inputs and argue that a suitable cost function to be imposed on the channel input is one that grows logarithmically. ii.We define novel and appropriate notions of power in such contexts. These notions boil down in the Gaussian context to the second moment which is the standard notion of power in the space of finite second moment variables. iii.In estimation theory, classical tools to quantify the estimator performance are tightly related to the assumption of a finite variance no