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 $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> . It demonstrates that the Tsallis and Sharma–Mittal entropies exhibit monotonic behavior with respect to $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , whereas two generalized forms of the Rényi entropy may exhibit “anomalous” (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $$\lambda \rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> and provide both lower and upper bounds for them.