Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior

TL;DR

This paper investigates how various entropy measures of the Poisson distribution change with its parameter, revealing monotonic and anomalous behaviors and providing bounds for large parameter values.

Contribution

It introduces a detailed analysis of entropy behaviors of the Poisson distribution, highlighting differences among entropy types and exploring asymptotic properties.

Findings

Tsallis and Sharma-Mittal entropies are monotonic in λ

Certain Rènyi entropies show non-monotonic 'anomalous' behavior

Bounds for entropies as λ approaches infinity are established

Abstract

The paper extends the analysis of the entropies of the Poisson distribution with parameter $\lambda$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $\lambda$, whereas two generalized forms of the R\'enyi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $\lambda \to \infty$ and provide both lower and upper bounds for them.