# Two Universality Properties Associated with the Monkey Model of Zipf's Law
## 2 Proof that \(-\beta\) Tends Towards \(-1\) from the Asymptotics of Log-Spacings  
For our analysis we specify a keyboard with an alphabet of \(K\geq 2\) letters \(\{L_{1}\ldots,L_{K}\}\) and a space character \(S\). The letter characters have non-zero probabilities \(q_{1},q_{2},\ldots,q_{K}\). The space character has probability \(s\), so that \(\sum_{i=1}^{K}q_{i}+s=1\). A word is defined as any sequence of non-space letters terminating with a space. A word \(W\) of exactly \(n\) letters is a string such as \(W=L_{i_{1}}L_{i_{2}}\ldots L_{i_{n}}S\) and has a probability of the form \(P(W)=P=q_{i_{1}}q_{i_{2}}\ldots q_{i_{n}}\)'s because letters are struck independently. The space character with no preceding letter character will be considered a word of length zero. The rank ordered sequence of descending word probabilities in the ensemble of all possible words is written \(P_{1}=s>P_{2}\geq\ldots P_{r}\geq\ldots\) (\(P_{1}=s\) is always the first and largest word probability). We break ties for words with equal probabilities by alphabetical ordering, so that each word probability has a unique rank \(r\). [PERSON] and [PERSON] [10] give a carefully constructed definition of power law behavior in the context of the monkey model as the situation where there exist \(C_{1},C_{2},\beta\) such that the inequality  
\[C_{1}r^{-\beta}\leq P_{r}\leq C_{2}r^{-\beta} \tag{1}\]  
holds for sufficiently large \(r\). It turns out that \(\beta>1\). A few comments about this definition are in order. First, note that there will always be many words with tied probability values. For example,the words \(L_{1}L_{2}S\) and \(L_{2}L_{1}S\) have the same probability \(q_{1}q_{2}s\). Because of this, the word probabilities in this model could never _exactly_ conform to a perfect inverse power law \(P_{r}=Cr^{-\beta}\). The asymptotic behavior for \(P_{r}\) is more subtle than this and highlights the significance of [PERSON] and [PERSON]'s analysis. They are able to give a stronger result when at least two letter probabilities have an irrational log-ratio \(\ln q_{i}/\ln q_{j}\). For this case, they show that there exists a constant \(C\) such that \(P_{r}\sim Cr^{-\beta}\), where \(\sim\) denotes asymptotic equivalence, \(\lim_{r\to\infty}P_{r}/(Cr^{-\beta})=1\). For our purposes here, we will simply refer to the [PERSON] and [PERSON] results as generally implying that \(P_{r}\) has _approximate power law behavior_. Note that inequality (1) implies the weaker asymptotic condition \(\ln P_{r}\sim-\beta\ln r\) as \(r\to\infty\). We'll call this _asymptotic log-log linear_ behavior and observe that on a log-log scale, \(\ln C_{1}\) and \(\ln C_{2}\) are asymptotically negligible so that only the exponent \(-\beta\) becomes significant. As can be seen in Figure 2b-d, to be explained shortly, monkey model word probabilities generated from unequal letter probabilities in the manner described below produce graphs distinctly log-log linear. [PERSON] and [PERSON] derive the exponent value \(-\beta\) but do not express it in what we consider its simplest form. Following [PERSON] and [PERSON] [11], the parameter \(\beta\) can be represented as the solution to  
\[q_{1}^{1/\beta}+q_{2}^{1/\beta}+\cdots+q_{K}^{1/\beta}=1, \tag{2}\]  
with \(\beta>1\). [PERSON] and [PERSON] [11] indicate \(\ln p(r)-\ln r/\gamma\) in the inequality for their Theorem 1, but what was evidently intended is \(\ln p(r)+\ln r/\gamma\). Their \(1/\gamma\) is equal to our \(\beta\). In the Miller model with equal letter probabilities, \(q_{1}=\cdots=q_{K}=(1-s)/K\), so from Equation (2), \(\beta\) in this case is found to be \(1-\ln(1-s)/\ln K\). In the Fibonacci example given by [PERSON] and [PERSON] [10] and in [PERSON] [17] they use \(K=2\) letters with probabilities \(q_{1}\), \(q_{2}=q_{1}^{2}\) and \(q_{1}<(-1+\sqrt{5})/2\) so that \(q_{1}+q_{2}<1\). Then \(\beta\) is the solution to \(q_{1}^{1/\beta}+q_{1}^{2/\beta}=1\), giving  
\[\beta = \frac{\ln q_{1}}{\ln((-1+\sqrt{5})/2)}. \tag{3}\]  
Conditions leading to \(-\beta\approx-1\) were not explored in [10; 11].