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2014 Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution
Yutao Ma, Zhengliang Zhang
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Electron. Commun. Probab. 19: 1-9 (2014). DOI: 10.1214/ECP.v19-3071

Abstract

In this paper, consider the circular Cauchy distribution $\mu_x$ on the unit circle $S$ with index $0\le |x|<1$, we study the spectral gap and the optimal logarithmic Sobolev constant for $\mu_x$, denoted respectively as $\lambda_1(\mu_x)$ and $C_{\mathrm{LS}}(\mu_x).$ We prove that $\frac{1}{1+|x|}\le \lambda_1(\mu_x)\le 1$ while $C_{\mathrm{LS}}(\mu_x)$ behaves like $\log(1+\frac{1}{1-|x|})$ as $|x|\to 1.$

Citation

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Yutao Ma. Zhengliang Zhang. "Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution." Electron. Commun. Probab. 19 1 - 9, 2014. https://doi.org/10.1214/ECP.v19-3071

Information

Accepted: 18 February 2014; Published: 2014
First available in Project Euclid: 7 June 2016

zbMATH: 1311.60029
MathSciNet: MR3167883
Digital Object Identifier: 10.1214/ECP.v19-3071

Subjects:
Primary: 60E15
Secondary: 26Dxx, ‎39B62

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