The exact power law and Pascal pyramid

Abstract

Let ${\omega }_0,{\omega }_1,\dots ,{\omega }_n$ be a full set of outcomes (letters, symbols) and let positive $p_i$, $i=0,\dots ,n$, be their probabilities (${\sum }_{i=0}^n{p}_i=1$). Let us treat ${\omega }_0$ 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 letters. We consider the list of all possible words sorted in the non-increasing order of their probabilities. Let $p(r)$ be the probability of the $r$th word in this list. We prove that if at least one of ratios $\log p_i/\log p_j$, $i,j\in \{1,\dots ,n\}$, is irrational, then the limit ${\lim }_{r\to \infty }p(r)/r^{1/\gamma }$ exists and differs from zero; here $\gamma$ is the root of the equation ${\sum }_{i=1}^n{p}_i^{\gamma }=1$. Some weaker results were established earlier. We are first to write an explicit formula for this limit constant at the power function; it can be expressed (rather easily) in terms of the entropy of the distribution~$(p_1^{\gamma },\dots ,p_n^{\gamma })$.

19 pages

References in corpus (2)

• Zipf and non-Zipf Laws for Homogeneous Markov Chain
• Two Universality Properties Associated with the Monkey Model of Zipf's Law