# Two Universality Properties Associated with the Monkey Model of Zipf's Law
## 4 Connections to Other Work
Modern studies of the _full_ distribution of communities, such as [PERSON]'s [28] analysis of the populations of 25,359 places from U.S. Census 2000, prove with high confidence that the _bulk_ of the community populations fit an approximate lognormal distribution and that the power law behavior is confined to the upper tail. [PERSON] and [PERSON] [29; 30] explained how to generate a Pareto-lognormal hybrid for income distributions by using a hierarchical mixture model of lognormal distributions. This is motivated from several angles, including the notion that higher classes amplify their income by organizing in such a way as to benefit from the efforts of others. [PERSON] and [PERSON] [31] have provided a far-ranging framework for understanding these hybrid distributions across the entire spectrum of disciplines where they have been discovered: \"physics, biology, geography, economics, insurance, lexicography, internet ecology, _etc._\" In [31] he and [PERSON] give a condensed summary showing that \"if stochastic processes with exponential growth in expectation are killed (or observed) randomly, the distribution of the killed or observed state exhibits power law behavior in one or both tails\". This work encompasses: (1) geometric Brownian motion (GBM); (2) discrete multiplicative processes; (3) homogeneous birth and death processes; and (4) Galton-Watson branching processes. In one of many variations on this theme, in his GBM model [32] [PERSON] specifies a stochastic process that uses lognormal distributions varying continuously in time with an exponential mixing distribution and shows that this generates a \"Double Pareto-Lognormal Distribution\". This has an asymptotic Pareto upper tail and a central part approximately lognormal; in addition, it exhibits interesting asymptotic behavior in the it lower tail where it is characterized by a direct (rather than an inverse) power law. [PERSON] has gone to great effort to demonstrate the high quality of the fit of this distribution for all ranges of values (low, middle, high) across a wide variety of size distributions such as particle and oil field sizes [32], incomes [33], internet file sizes and the sizes of biological general [31]. In today's nomenclature, the term _Zipf's law_ has come to mean any empirical distribution with an approximate power law form and an exponent close to the value \(-1\), not just the word frequency law. (To be clear here, when we refer to power law distributions, we also mean the hybrids with power law tails that we have been considering.) The subset of power laws with this restricted exponent value is surprisingly large and includes the distribution of firm sizes [34], city sizes [35], the famous Gutenberg-Richter law for earthquake magnitudes [36] and many others. [PERSON] [35] pointed out that while it has long been known that stochastic growth processes could generate power laws, \"these studies stopped short of explaining why the Zipf exponent should be 1.\" To address this question, he has given a theoretical explanation of the genesis of a \(-1\) exponent for the Zipf law for city populations, but in fact, his approach can be regarded more generally. His key idea is a variation of [PERSON]'s [37]_Law of Proportion Effect_: using a fixed number of normalized city population quantities that sum to 1 and assuming growth processes (expressed as percentages) randomly distributed as i.i.d. variables with a common mean and variance, he proves that a steady state will satisfy [PERSON]'s law. This proof requires an additional strong assumption in order to reach a steady state, namely, a lower bound for the (normalized) population size. [PERSON] goes on to discuss relaxed versions of his model and how it fits into the larger context of similar work done by others. In their monograph on [PERSON]'s law [PERSON] _et al._[38] modify and extend the core [PERSON] idea in numerous ways that render it more realistic.