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

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.