# Two Universality Properties Associated with the Monkey Model of Zipf's Law
## 1 Introduction  
In his popular expository article on universality examples in mathematics and physics [PERSON] [1] mentions [PERSON]'s empirical power law of word frequencies as lacking any \"convincing explanation for how the law comes about and why it is universal\". By _universality_ he means the idea where systems follow certain macroscopic laws that are largely independent of their microscopic details. In this article we drop down many levels from natural language to discuss _two_ universality properties associated with the toy monkey model of [PERSON]'s law. Both of these properties were previously considered in [PERSON] [2], but the presentation there was incompletely developed, and here we clarify and greatly expand upon these two ideas by showing: (1) how the model displays a nearly universal tendency towards a \(-1\) exponent in its approximate power law behavior; and (2) the significance of the central limit theorem (CLT) for a random number of i.i.d. variables in the model with a finite word length cutoff. We will sometimes refer to the case with a finite word length cutoff as _Monkey Twitter_, as explained in Section 3. Our analysis regarding the very general tendency towards a \(-1\) exponent when letter probabilities are drawn as a sample from any of an enormous class of probability distributions is a new result in the literature on this topic. Our application of the CLT with a random number of summands was first introduced with respect to the monkey model in [2], and we want to emphasize here that this is a significant aspect of the structure of the model. Our paper is laid out as follows. In Section 2 we prove the strong tendency towards an approximate \(-1\) exponent under very broad conditions by means of a limit theorem for the logarithmsof random spacing due to [PERSON] and [PERSON] [3]. An equivalent but relatively longer proof of this, still based on [3], has been given in [4]. In Section 3, we explain the underlying lognormal structure of the _central part_ of the distribution of word probabilities in the case of a _finite word length_ cutoff and show how it leads to a hybrid Zipf-lognormal distribution (called a lognormal-Pareto distribution in [2]). These are universality properties in exactly [PERSON]'s sense because the distribution of the word probabilities (the macro behavior) is within very broad bounds independent of the details of how the letter probabilities for the monkey's typewriter are selected (the micro behavior). In Section 4, we discuss the multiple connections between these results and well-known research from other areas where Pareto-Zipf type distributions have been investigated. In the remainder of this section we sketch some historical background. It has been known for many years that the monkey-at-the-typewriter scheme for generating random \"words\" conforms approximately to an inverse power law of word frequencies and therefore roughly mimics the approximate inverse power form of [PERSON]'s [5] statistical regularity for many natural languages [6]. _However, [PERSON]'s empirical word frequency law not only approximates a power distribution, but as he emphasized, it also very frequently exhibits an exponent in the neighborhood of \(-1\)_. That is, letting \(f_{1}\geq f_{2}\geq\ldots f_{r}\cdots\geq\ldots\) represent the observed ranked word frequencies, [PERSON] found \(f_{r}\approx Cr^{-1}\), with \(C>0\) a constant depending on the sample size. Many qualifications have been raised regarding this approximation (including the issue of the divergence of the harmonic series); yet as [PERSON] [7] commented: \"The Zipf law is unreliable but it is often a good enough approximation to demand an explanation\". Figure 1 illustrates the fascinating universal character of this word frequency law using the text from four different authors writing in four different European languages in four different centuries. The plots are shown in the log-log form that [PERSON] employed, and they resemble countless other examples that researchers have studied since his time.