  The distribution of word probabilities in the monkey model of Zipf's law is
associated with two universality properties: (1) the power law exponent
converges strongly to $-1$ as the alphabet size increases and the letter
probabilities are specified as the spacings from a random division of the unit
interval for any distribution with a bounded density function on $[0,1]$; and
(2), on a logarithmic scale the version of the model with a finite word length
cutoff and unequal letter probabilities is approximately normally distributed
in the part of the distribution away from the tails. The first property is
proved using a remarkably general limit theorem for the logarithm of sample
spacings from Shao and Hahn, and the second property follows from Anscombe's
central limit theorem for a random number of i.i.d. random variables. The
finite word length model leads to a hybrid Zipf-lognormal mixture distribution
closely related to work in other areas.

Probability;Physics and Society