# Two Universality Properties Associated with the Monkey Model of Zipf's Law
## 4 Connections to Other Work  
The subject of power law distributions is a vast topic of research reaching across diverse scientific disciplines [17; 22; 23; 24]. Here we provide a brief sketch of how our results connect to some other work and how they can be considered in a much more general light. The monkey model viewed in terms of its hierarchical tree structure is the starting point for understanding its general nature. Though [PERSON] did not explicitly present his model as a branching tree, several researchers at almost the same time in the mid-to-late 1950s were highlighting this form using the equivalent of equal letter probabilities to motivate the occurrence of empirical power law distribution in other areas. For example, [PERSON] [25] introduced this idea in connection with the power law of city populations within a country (\"Auerbach's law\" [26]). Using essentially the same logic as [PERSON], but in a completely different setting, [PERSON] assumed that a given community will have \(K\) satellite communities, each with a constant decreasing fraction of the population at a higher level. That is, if \(A\) is the population of the largest city and \(0<p<1\), there will be \(K\) nearby satellite communities with population \(pA\), \(K^{2}\) smaller communities nearby to these with population \(p^{2}A\), and so on. [PERSON] (p. 226, [27]) had an apt expression for this pattern: he described it as \"compensation between two exponentials\" because the number of observations at each level increases geometrically while the mean value decreases geometrically. [PERSON] then went on to note that if instead of using a constant decreasing fraction \(p\), one used a random variable \(X\) on \((0,1)\), the population at the \(n^{\text{th}}\) level down would be a random variable of the form \(X_{1}X_{2}\cdots X_{n}A\), leading to an approximate lognormal distribution of populations at this level. This corresponds to our discussion of monkey word probabilities in \(\mathbf{P}_{\text{len}=n}\), although [PERSON] was not aware of the still stronger statement of lognormality for the probabilities of words of length \(\leq n\) in \(\mathbf{P}_{\text{len}\leq n}\) that we demonstrated using [PERSON]'s CLT. Many more examples of a branching tree structure essentially equivalent to [PERSON]'s monkey model with equiprobable letters have been proposed over the years to motivate the occurrence of a huge variety of empirical power law distributions, including such size distributions as lake areas, island areas (\"Korcak's law\"), river lengths, etc. Indeed, [PERSON]'s [18] classic book on fractals is a rich source of these. This equiprobability case is so simple to analyze that it has been invoked over and over again to illustrate how power law behavior can arise. However, the more complicated case using unequal letter probabilities (or proportions) _and a finite word length cutoff_ is what really underscores the close analogy between the monkey Zipf-lognormal distribution and other mixture models exhibiting hybrid power law tails and approximately lognormal central ranges. Empirical distributions of this form have appeared with great frequency. In fact there is a long history of researchers, including [PERSON], first discovering what appears to be an empirical power law over an entire distribution because they start out by looking at only the largest observations (cities, corporations, islands, _etc._). However, when they extend their measurements to the part of the distribution below the upper tail, _which is always more difficult to observe,_ the power law behavior typically breaks down. Several examples of this are chronicled in [22]. The power law for city sizes noted above is a perfect illustration. [PERSON] in 1913 [26] looked at the 94 largest German cities from a 1910 census and showed a good power law fit, but because he did not include the thousands of smaller communities, he may not have been aware that this relationship breaks down.