Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

Yutao Ma (Beijing Normal University)
Zhengliang Zhang (Wuhan University)


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 $|x|\to 1.$

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Pages: 1-9

Publication Date: February 18, 2014

DOI: 10.1214/ECP.v19-3071


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