I'd like to have a reference for the property $0 \leq \alpha < \alpha' \leq \infty \implies R_\alpha(\mu) > R_{\alpha'}(\mu)$, where $R_\alpha(\mu)$ is the Rényi entropy of order $\alpha$ of a probability $\mu$ on a finite set. The book Thermodynamics of chaotic systems: an introduction by Beck & Schlögl provides a proof of this inequality but does not provide the equality $\lim_{\alpha\to\infty} R_\alpha(\mu) = -\log \max \{\mu_i\} =: R_\infty(\mu)$.
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2 Answers
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There is the simple inequality $$ (\max\mu_i)^\alpha\le \sum_i {\mu_i^\alpha}\le k(\max\mu_i)^\alpha, $$ where $k$ is the size of the set. Taking logarithms and dividing by $1-\alpha$, one sees that $R_\alpha(\mu)$ is bounded above and below by quantities that converge to $-\log\max \mu_i$ as $\alpha\to\infty$.
answered Jun 14, 2016 at 22:08
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$\begingroup$ Ok, thank you. I was looking for a reference, not a proof. But it's true that the proof is very easy, so that a reference is not essential. $\endgroup$ Commented Jun 17, 2016 at 19:50
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This follows from the consideration of the power mean $$M_{a}(p_1,\cdots,p_n)=\left(\frac{\sum_{i=1}^n p_i^a}{n}\right)^{1/a}$$ which is monotone increasing and tends to $$\max \{p_i:1\leq i\leq n\}$$ as $a \rightarrow \infty.$ See, p.74 onwards in the book Analytic Inequalities by Mitrinovic.
