Random texts exhibit Zipf's-law-like word frequency distribution

It is shown that the distribution of word frequencies for randomly generated texts is very similar to Zipf's law observed in natural languages such as the English. The facts that the frequency of occurrence of a word is almost an inverse power law function of its rank and the exponent of this inverse power law is very close to 1 are largely due to the transformation from the word's length to its rank, which stretches an exponential function to a power law function. Zipf observed long time ago [1] that the distribution of word frequencies in English, if the words are aligned according to their ranks, is an inverse power law with the exponent very close to 1. In other words, if the most frequently occurring word appears in the text with the frequency P(l), the next most frequently occurring word has the frequency P(2), and the rank-r word has the frequency P(r), the frequency distribution is C P(r) =-, r" (0.1 ) with C R:i 0.1 and Ct R:i 1. This distribution, also called Zipf's law, has been checked for accuracy for the standard corpus of the present-day English with very good results [2]. The fall-off of the distribution as the rank is increased is obvious, because the more frequently occurring words are guaranteed to have larger frequencies than those less frequently occurring. Nevertheless, it seems to be a puzzle as why the decay is a power law instead of an exponential function or other faster decaying functions, and why the exponent is very close to 1 instead of 2 or even larger values. There are attempts to incorporate Zipf's law into the grander framework of "fractals" [3], but in doing so, little insight has been gained in understanding this particular "law." Probably few people pay attention to a comment by Miller in his preface to Zipf's book [4], that randomly generated texts, which are perhaps the least interesting sequences and unrelated to any other scaling behaviors, also exhibit Zipf's law. What he said was that Zipf's law is not exclusive for English or any other natural languages. Miller did not give