engleski

Zipf-Mandelbrot law, f-divergences and the Jensen-type interpolating inequalities

The paper is motivated by the method of interpolating inequalities that makes use of the improved Jensen-type inequalities. This approach is integrated with the well known Zipf-Mandelbrot law applied to various types of f-divergences, such are Kullback-Leibler divergence, Hellinger distance, Bhattacharyya distance (via coefficient), chi-square divergence, total variation distance and triangular discrimination. General results are firstly deduced for the Csiszar divergence functional from which the listed divergences originate. The analyzed inequalities are presented for the ZipfMandelbrot law and then accentuated for its special form – the Zipf law with its special role in linguistics. This aspect is introduced through the Zipfian word distribution associated to the English and Russian languages, using the obtained bounds for the Kullback-Leibler divergence. Thus its experimental character is accentuated. Since this paper includes three important and widely investigated issues: the Jensen inequality, the divergences for probability distributions and the Zipf-Mandelbrot law, it can be of interest for mathematicians who investigate any of these fields with an accent put on mathematical inequalities, as well as for interdisciplinary fields (e.g. linguistics was involved in this case.)

Csiszar divergence functional, f-divergences, Jensen inequality, Kullback-Leibler divergence, Zipf and Zipf-Mandelbrot law.

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