Range-Renewal Processes: SLLN, Power Law and Beyonds

TL;DR

This paper investigates the asymptotic behavior of samples from discrete distributions, establishing strong laws of large numbers, power law phenomena, and properties of graphs derived from samples, with implications for small-world networks.

Contribution

It introduces new strong law results for discrete distributions, links heavy tails to power laws, and analyzes graph degree distributions and small-world phenomena.

Findings

Proves strong law of large numbers for discrete distributions.

Shows power law emergence in heavy-tailed distributions.

Analyzes degree distribution and small-world properties in constructed graphs.

Abstract

Given $n$ samples of a regular discrete distribution $\pi$, we prove in this article first a serial of SLLNs results (of Dvoretzky and Erd\"{o}s' type) which implies a typical power law when $\pi$ is heavy-tailed. Constructing a (random) graph from the ordered $n$ samples, we can establish other laws for the degree-distribution of the graph. The phenomena of small world is also discussed.