## Tbilisi Mathematical Journal

- Tbilisi Math. J.
- Volume 11, Issue 3 (2018), 1-27.

### Majorizatiuon and Zipf-Mandelbrot law

Naveed Latif, Đilda Pečarić, and Josip Pečarić

#### Abstract

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 $\alpha$-order entropy for Z-M law, variational distance for Z-M law, the Hellinger distance for Z-M law, $\chi^{2}$-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.

#### Note

The publication was supported by the Ministry of Education and Science of the Russian Federation (the Agreement number No. 02.a03.21.0008.) This publication is partially supported by Royal Commission for Jubail and Yanbu, Kingdom of Saudi Arabia.

#### Article information

**Source**

Tbilisi Math. J., Volume 11, Issue 3 (2018), 1-27.

**Dates**

Received: 22 July 2017

Accepted: 21 February 2018

First available in Project Euclid: 3 October 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.tbilisi/1538532023

**Digital Object Identifier**

doi:10.32513/tbilisi/1538532023

**Mathematical Reviews number (MathSciNet)**

MR3757497

**Zentralblatt MATH identifier**

1369.94404

**Subjects**

Primary: 94A15: Information theory, general [See also 62B10, 81P94]

Secondary: 94A17: Measures of information, entropy 26A51: Convexity, generalizations 26D15: Inequalities for sums, series and integrals

**Keywords**

Majorization inequailty Csiszár $f$-divergence Zipf-Mandelbrot law Zipf's law in linguistic Rényi $\alpha$-order entropy variational distance Hellinger discrimination $\chi^{2}$-distance and triangular discrimination

#### Citation

Latif, Naveed; Pečarić, Đilda; Pečarić, Josip. Majorizatiuon and Zipf-Mandelbrot law. Tbilisi Math. J. 11 (2018), no. 3, 1--27. doi:10.32513/tbilisi/1538532023. https://projecteuclid.org/euclid.tbilisi/1538532023