Majorizatiuon and Zipf-Mandelbrot law
N. Latif, Ð. Pečarić,
In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the
Csiszár f-divergence for the Zipf-Mandelbrot law and then develop important
majorization inequalities for these divergences. We also discuss some special cases for our generalized results by using the Zipf-Mandelbrot law.
As applications, we present the majorization inequalities for various distances obtaining by some special convex
functions in the Csiszár f-divergence for Z-M law like the
Rényi α-order entropy
for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, χ2-distance for Z-M law
and triangular discrimination for Z-M law. At the end, we give important applications of the Zipf's
law in linguistics and obtain the bounds for the Kullback-Leibler divergence of the distributions
associated to the English and the Russian languages.
Tbilisi Mathematical Journal, Vol. 11(3) (2018), pp. 1-27