Majorizatiuon and Zipf-Mandelbrot law

N. Latif, . Pečarić, J. Pečarić

In this paper we show how the Zipf-Mandelbrot law is connected to the theory of majorization. Firstly we consider the Csiszr 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 Csiszr f-divergence for Z-M law like the Rnyi 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