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Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

   
@article{ECP3071,
	author = {Yutao Ma and Zhengliang Zhang},
	title = {Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution},
	journal = {Electron. Commun. Probab.},
	fjournal = {Electronic Communications in Probability},
	volume = {19},
	year = {2014},
	keywords = {circular Cauchy distribution, spectral gap,logarithmic Sobolev inequalities},
	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 $|x|\to 1.$},
	pages = {no. 10, 1-9},
	issn = {1083-589X},
	doi = {10.1214/ECP.v19-3071},    
        url = {http://ecp.ejpecp.org/article/view/3071}}