Calculation of Precise Constants in a Probability Model of Zipf’s Law Generation and Asymptotics of Sums of Multinomial Coefficients

Let be a full set of outcomes (symbols) and let positive , , be their probabilities . Let us treat as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the nonincreasing order of their probabilities. Let be the probability of the th word in this list. We prove that if at least one of the ratios , , is irrational, then the limit exists and differs from zero; here is the root of the equation . The limit constant can be expressed (rather easily) in terms of the entropy of the distribution .

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