Range-Renewal Processes: SLLN, Power Law and Beyonds

Given $n$ samples of a regular discrete distribution $π$, we prove in this article first a serial of SLLNs results (of Dvoretzky and Erdös' type) which implies a typical power law when $π$ 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.

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