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<p>The paper extends the analysis of the entropies of the Poisson distribution with parameter&#xa0;<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10171_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. It demonstrates that the Tsallis and Sharma–Mittal entropies exhibit monotonic behavior with respect to <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10171_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, whereas two generalized forms of the Rényi entropy may exhibit “anomalous” (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11009_2025_10171_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \rightarrow \infty \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> and provide both lower and upper bounds for them.</p>

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Entropies of the Poisson Distribution as Functions of Intensity: “Normal” and “Anomalous” Behavior

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月22日

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  • Dmitri Finkelshtein,
  • Anatoliy Malyarenko,
  • Yuliya Mishura,
  • Kostiantyn Ralchenko

摘要

The paper extends the analysis of the entropies of the Poisson distribution with parameter  \(\lambda \) λ . It demonstrates that the Tsallis and Sharma–Mittal entropies exhibit monotonic behavior with respect to \(\lambda \) λ , whereas two generalized forms of the Rényi entropy may exhibit “anomalous” (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as \(\lambda \rightarrow \infty \) λ → ∞ and provide both lower and upper bounds for them.