Range-Renewal Processes: SLLNs and Power Laws

Given n samples (viewed as an n-tuple) of a γ-regular discrete distribution π, in this article the authors concern with the weighted and unweighted graphs induced by the n samples. They first prove a series of SLLN results (of Dvoretzky-Erdös’ type). Then they show that the vertex weights of the graphs under investigation obey asymptotically power law distributions with exponent 1 + γ. They also give a conjecture that the degrees of unweighted graphs would exhibit asymptotically power law distributions with constant exponent 2. This exponent is obviously independent of the parameter γ ∈ (0, 1), which is a surprise to us at first sight.