Heavy Tailed Distributions in Telecommunications: Statistical Modelling through Alpha-Theory

In this Master thesis we will focus on the peculiar statistical properties of heavy tailed distributions and we will show some of their important engineering applications in the telecommunications world. In practice, the most relevant heavy tailed distributions are those with finite mean and divergent variance. The LogNormal distribution, although it appears to be an accurate model for the data, is often discarded for modelling heavy tailed phenomena because, in its standard version, it has finite variance. Using the main theoretical foundations of Alpha-Theory, an advancement in Non-Standard Analysis, introduced by Vieri Benci in late nineties, we will show how to create an Euclidean version of the LogNormal distribution, with well-defined infinite variance (i.e. a numerical value, not a potential diverging infinity, as expressed by the symbol lemniscate). Through experimental tests in Matlab, we will also be able to generate samples that follow this distribution and subsequently verify numerically that the sample values of mean and variance are consistent with the theoretical values, obtaining a convincing result