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September 2018 Majorizatiuon and Zipf-Mandelbrot law
Naveed Latif, Đilda Pečarić, Josip Pečarić
Tbilisi Math. J. 11(3): 1-27 (September 2018). DOI: 10.32513/tbilisi/1538532023

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.

Funding Statement

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.

Citation

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Naveed Latif. Đilda Pečarić. Josip Pečarić. "Majorizatiuon and Zipf-Mandelbrot law." Tbilisi Math. J. 11 (3) 1 - 27, September 2018. https://doi.org/10.32513/tbilisi/1538532023

Information

Received: 22 July 2017; Accepted: 21 February 2018; Published: September 2018
First available in Project Euclid: 3 October 2018

zbMATH: 1369.94404
MathSciNet: MR3757497
Digital Object Identifier: 10.32513/tbilisi/1538532023

Subjects:
Primary: 94A15
Secondary: 26A51‎, 26D15, 94A17

Rights: Copyright © 2018 Tbilisi Centre for Mathematical Sciences

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Vol.11 • No. 3 • September 2018
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