# Statistical Divergence and Paths Thereof to Socioeconomic Inequality and to Renewal Processes

## TL;DR

This paper introduces a new way to measure statistical differences and applies it to topics like inequality and renewal processes.

## Contribution

A general framework for statistical divergence and its novel applications to socioeconomic inequality and renewal processes.

## Key findings

- The framework relates to known measures like f-divergence and Kullback–Leibler divergence.
- It connects statistical divergence to socioeconomic inequality indices like the Gini index.
- It provides new insights into the divergence of renewal processes from Poisson processes.

## Abstract

This paper establishes a general framework for measuring statistical divergence. Namely, with regard to a pair of random variables that share a common range of values: quantifying the distance of the statistical distribution of one random variable from that of the other. The general framework is then applied to the topics of socioeconomic inequality and renewal processes. The general framework and its applications are shown to yield and to relate to the following: f-divergence, Hellinger divergence, Renyi divergence, and Kullback–Leibler divergence (also known as relative entropy); the Lorenz curve and socioeconomic inequality indices; the Gini index and its generalizations; the divergence of renewal processes from the Poisson process; and the divergence of anomalous relaxation from regular relaxation. Presenting a ‘fresh’ perspective on statistical divergence, this paper offers its readers a simple and transparent construction of statistical-divergence gauges, as well as novel paths that lead from statistical divergence to the aforementioned topics.