# Two Universality Properties Associated with the Monkey Model of Zipf's Law
## 3 Monkey Twitter: [PERSON]'s CLT for the Model with a Finite Word Length Cutoff
The plot in Figure 3a was generated just as in Figure 2b except that a finite length cutoff of \(n\leq 4\) letters has been applied. To make this clear, in Figure 2b the plot shows the top 475,255 word probabilities in \(\mathbf{P}_{\infty}\) generated using \(K=26\) letters derived from uniform spacings. In contrast, using the same letter probabilities, the plot in Figure 3a shows the 475,255 word probabilities generated with word length \(\leq 4\) letters, _i.e._, all the values of \(P\in\mathbf{P}_{\text{len}\leq 4}\). Word length in the case of _unequal_ letter probabilities is certainly correlated, _but not perfectly_, with word probability: words of shorter length will, _on average_, have a higher probability than those of longer length, but except in [PERSON]'s degenerate case, there will always be reversals where a longer word will have a higher probability than a shorter word. In natural languages, [PERSON] [5] underscored that \"the length of a word tends to bear an inverse relationship to its relative frequency\", which he called the _Law of Abbreviation_. In many respects, this is the starting point for his _Principle of Least Effort_. However, writing \(P_{1:N_{n}}>P_{2:N_{n}}\cdots\geq P_{N_{n}:N_{n}}\) for the ranked values of \(\mathbf{P}_{\text{len}\leq\text{n}}\), it should be evident that for any rank \(r\), \(P_{r:N_{n}}\leq P_{r}\) and that \(P_{r:N_{n}}=P_{r}\) when \(n\) is sufficiently large. In short, \(\mathbf{P}_{\text{len}\leq\text{n}}\) inherits its upper tail approximate power law behavior from \(\mathbf{P}_{\infty}\), which is illustrated by comparing the top part of the curve in Figure 3a with the corresponding part of Figure 2b. Figure 3: Figure 3a is a log-log plot of monkey word probabilities by rank for all words of length \(\leq 4\) non-space characters using letter probabilities from uniform spacings. The linear upper tail coincides closely with the previous Figure 2b, but the approximate power law clearly breaks down. Figure 3b shows the same word probabilities in a standard normal quantile plot. Its rough linearity confirms an approximate Gaussian fit over the whole distribution even though the upper tail is an approximate power law as seen on the left in Figure 3a. We refer to this distribution as a Zipf-lognormal hybrid, and it has many connections to other distributions discussed in the statistical literature. Figure 3b uses the same data points from Figure 3a graphed as a normal quantile plot, and its roughly linear appearance for the logarithm of word probabilities for \(P\in\mathbf{P}_{\text{len}\leq n}\) conforms to an approximate Guassian fit, although certainly the bending in the upper half of the distribution departs a bit from the linear trend of the lower half. The fact that a distribution can have a power law tail and lognormal central part and _still look lognormal over essentially its entire range_ may seem surprising. The discussion in [22] about _power law mimicry_ in the upper tail of lognormal distributions helps to explain why this can happen. We will call the distribution for \(\mathbf{P}_{\text{len}\leq n}\) a Zipf-lognormal distribution. The monkey model with a fixed word length cutoff can prove useful as a motivating idea. In the next section, we will discuss how models with the same branching tree structure have been proposed many times in the past, typically in settings where something like a finite word length is natural to consider. For the moment, think of the social networking service, Twitter, which allows members to exchange messages limited to at most 140 characters. Define _Monkey Twitter_ with a finite limit of \(n+1\) characters. For convenience, in any implementation of a Monkey Twitter random experiment we will assume that monkeys would always fill up their allotted message space of \(n+1\) characters. Monkey words still require a terminating space character, so it is possible for a monkey to type (at most) one \"non-word\", which can vary in length from 1 to \(n+1\) characters, and will always be the last part of a message string.