User aaron sheldon - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T15:13:38Z http://mathoverflow.net/feeds/user/12302 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/39224/is-there-a-natural-random-process-that-is-rigorously-known-to-produce-zipfs-law/52441#52441 Answer by Aaron Sheldon for Is there a natural random process that is rigorously known to produce Zipf's law? Aaron Sheldon 2011-01-18T22:37:51Z 2011-01-18T22:44:26Z <p>I wonder if there is a Zipf's Law equivalent of the Glivenko-Cantelli Theorem, where the condition of independence is replaced by requiring a sum of powers remain fixed, and the identical assumption is replaced by a marginal identical assumption?</p> <p>If $C=\sum_{i=1}^n X_i^\alpha$ and $X_i \sim P(X)$ then $\lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i=1}^{n} I(X_i \leq x) \propto \ln(x)$ almost surely?</p> <p>I ask this because empirically Zipf's Law seems to occur when the sample violates independence, and is usually constrained by requiring a fixed sum (like fixed population size, wealth, or resource). </p>