# 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 discussion of the finite word length version of the monkey model in [2] was incomplete because it did not explicitly provide the normalizing constants for the application of [PERSON]'s CLT for a random number of i.i.d. variables, as we do here. We also want to explain more clearly the nature of the hybrid Zipf-lognormal distribution that results from a finite length cutoff when letter probabilities are not identical. The focus in Section 2 on the infinite ensemble of word probabilities \(\mathbf{P}_{\infty}\)_ranked_ to give \(P_{1}>P_{2}\geq P_{3}\dots\) has actually obscured the significant _hierarchical structure_ of the monkey  
Figure 2: Log-log plots of monkey word probabilities by rank showing the asymptotic tendency towards a \(-1\) exponent (slope on the log-log scales) using four different distributions to generate letter probabilities: equal probabilities in (**a**) and a generalized broken stick process in (**b–d**). \(K=26\) letters were used in all cases and the largest \(\sum_{i=0}^{4}26^{i}=475,255\) word probabilities are displayed. model--what [PERSON] (p. 345, [18]), called a _lexicographic tree_--which only becomes evident when word length is considered. To see this, write \(\mathbf{P}_{\mathrm{len}=n}\) for the multiset of all word probabilities for monkey words of exactly length \(n\). There are \(K^{n}\) probabilities in \(\mathbf{P}_{\mathrm{len}=n}\) having a total sum of \((q_{1}+q_{2}+\cdots+q_{K})^{n}s\,=\,(1-s)^{n}s\) and an average value of \(((1-s)/K)^{n}s\), declining geometrically with \(n\). Now taking word length into account, [PERSON]'s lexicographic tree structure becomes clear with the simple case of \(K=2\) letters: the root node of the tree has a probability of \(s\) (for the \"space\" word of length \(0\)); it has two branches to the next level consisting of nodes for words of length \(1\) with probabilities \(q_{1}s\) and \(q_{2}s\); these each branch out to the next level with four nodes having probability values \(q_{15}^{2},q_{1}q_{2}s,q_{2}q_{1}s,q_{2}^{2}s\) and so on. The formation of word probabilities as the product of letter probabilities immediately suggests lognormal structure (normal on a log scale). A simple way to exhibit the underlying approximate lognormal character of the finite multisets of word probabilities for all words of exactly length \(n\) and those of length \(\leq\,n\) is to construct their natural product probability spaces using the basic counting measure for letter probability values. For this analysis it makes no difference how the letter probabilities \(q_{i}\) have been specified. We only require that they are positive values and at least two of them are different from each other. The relevance of the CLT is first seen by representing any probability \(P\in\mathbf{P}_{\mathrm{len}=n}\) as a product of i.i.d. random variables times the constant \(s\): \(P=X_{1}X_{2}\ldots X_{n}s\), where each \(X_{i}\) takes on one of the letter probability values \(q_{1},q_{2},\ldots,q_{K}\) with probability \(1/K\) (_i.e._, we use the natural counting measure to construct a probability space on \(\mathbf{P}_{\mathrm{len}=n}\)). Let \(\mu_{1}=\sum_{i=1}^{K}\ln q_{i}/K\) and \(\sigma_{1}^{2}=\sum_{i=1}^{K}(\ln q_{i}-\mu_{1})^{2}/K\). Assume that \(\sigma_{1}^{2}>0\), _i.e._, the letter probabilities are not all equal. Then \(\mu_{1}+\ln s\) is the mean and \(\sigma_{1}^{2}\) is the variance of \(\ln P\) for \(P\in\mathbf{P}_{\mathrm{len}=1}\), and it follows that \(n\mu_{1}+\ln s\) and \(n\sigma_{1}^{2}\) are the respective mean and variance of \(\ln P=S_{n}=\sum_{i=1}^{n}\ln X_{i}+\ln s\) for \(P\in\mathbf{P}_{\mathrm{len}=n}\). Therefore, \((S_{n}-n\mu_{1})/\sqrt{n\sigma_{1}^{2}}\) is asymptotically normally distributed \(N(0,1)\) (the term \(\ln s\) is rendered negligible in the asymptotics) so that for sufficiently large \(n\), \(P\) itself will be approximately lognormal \(LN(n\mu_{1},n\sigma_{1}^{2})\).