Properties of Shannon and Rényi entropies of the Poisson distribution as the functions of intensity parameter

Abstract

We consider two types of entropy, namely, Shannon and Rényi entropies of the Poisson distribution, and establish their properties as the functions of intensity parameter. More precisely, we prove that both entropies increase with intensity. While for Shannon entropy the proof is comparatively simple, for Rényi entropy, which depends on additional parameter $α>0$, we can characterize it as nontrivial. The proof is based on application of Karamata's inequality to the terms of Poisson distribution.

Figures (8)

• Figure 1: $\psi(\alpha_0,\lambda)$ is an increasing function of $\lambda$ when $\alpha_0$ is fixed, $0<\alpha<1$, $\lambda>0$.
• Figure 2: $\psi(\alpha,\lambda)$ is an increasing function of $\lambda$, $0<\alpha<1$, $\lambda>0$.
• Figure 3: $\psi(\alpha_0,\lambda)$ is an decreasing function of $\lambda$ when $\alpha_0$ is fixed, $\alpha>1$, $\lambda>0$.
• Figure 4: $\psi(\alpha,\lambda)$ is an decreasing function of $\lambda$, $\alpha>1$, $\lambda>0$.
• Figure 5: $R(\alpha_0,\lambda)>0$ when $0<\alpha_0<1$ is fixed, $\lambda>0$.

Theorems & Definitions (12)

• Theorem 1
• proof
• Lemma 1
• proof
• Theorem 2
• proof
• Lemma 2
• proof
• Remark 1
• Lemma 3