Is the Gaussian distribution "Normal"? Signal Processing with Alpha-Stable Distributions

There are solid reasons for the popularity of Gaussian models. They are easy to deal with, lead to linear equations, and they have a strong theoretical justification given by the Central Limit theorem. However, many data, manmade or natural, exhibit characteristics too impulsive or skewed to be successfully accommodated by the Gaussian model. The wide spread power laws in the nature, in internet, in linguistics, biology are very well known. In this talk we will challenge the "Normality" of the Gaussian distribution and will discuss the alpha‐stable distribution family which satisfies the generalised Central Limit Theorem. Alpha‐Stable distributions have received wide interest in the signal processing community and became state of the art models for impulsive noise and internet traffic in the last 20 years since the influential paper of Nikias and Shao in 1993. We will provide the fundamental theory and discuss the rich class of statistics this family enables us to work with including fractional order statistics, log statistics and extreme value statistics. We will present some application areas where alpha‐stable distributions had important success such as internet traffic modelling, SAR imaging, computational biology, astronomy, etc. We will identify open problems which we hope will lead to fruitful discussion on further research on this family of distributions.