On the Sum of Random Samples with Bounded Pareto Distribution

TL;DR

This paper derives the distribution of the sum of bounded Pareto random variables, addressing practical limitations of heavy-tailed models by incorporating truncation and censoring, with analytical and numerical insights.

Contribution

It introduces new analytical expressions for the sum of truncated and censored Pareto Type-II variables, extending traditional Pareto models to more realistic bounded scenarios.

Findings

Derived the distribution of the sum of right-censored Pareto Type-II variables.

Connected the truncated Pareto sum distribution with the unbounded case.

Provided numerical analysis illustrating the mixture structure of the distributions.

Abstract

Heavy-tailed random samples, as well as their sum or average, are encountered in a number of signal processing applications in radar, communications, finance, and natural sciences. Modeling such data through the Pareto distribution is particularly attractive due to its simple analytical form, but may lead to infinite variance and/or mean, which is not physically plausible: in fact, samples are always bounded in practice, namely because of clipping during the signal acquisition or deliberate censoring or trimming (truncation) at the processing stage. Based on this motivation, the paper derives and analyzes the distribution of the sum of right-censored Pareto Type-II variables, which generalizes the conventional Pareto (Type-I) and Lomax distributions. The distribution of the sum of truncated Pareto is also obtained, and an analytical connection is drawn with the unbounded case. A numerical analysis illustrates the findings, providing insights on several aspects including the intimate mixture structure of the obtained expressions.