Inequalities for indices of coincidence and entropies

TL;DR

This paper explores inequalities and recurrence relations for the index of coincidence, which are used to analyze the Rényi and Tsallis entropies of probability distributions depending on a parameter.

Contribution

It introduces new recurrence relations and inequalities for the index of coincidence to better understand associated entropies.

Findings

Derived inequalities for the index of coincidence

Established recurrence relations for S(x)

Provided tools to analyze Rényi and Tsallis entropies

Abstract

We consider a probability distribution depending on a real parameter $x$. As functions of $x$, the R\'enyi entropy and the Tsallis entropy can be expressed in terms of the associated index of coincidence $S(x)$. We establish recurrence relations and inequalities for $S(x),$ which can be used in order to get information concerning the two entropies.