Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution
Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution
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DOI：
10.1214/ecp.v19-3071
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发表时间：
2014-02
影响因子：
0.5
通讯作者：
Yutao Ma;Zheng-liang Zhang
中科院分区：
文献类型：
--
作者：
Yutao Ma;Zheng-liang Zhang
本文考虑单位圆S上指标为0\le的圆柯西分布|X| <1$，我们研究了$\mu_x$的谱隙和最佳对数Sobolev常数，分别记为$\mathrm {LS}（\mu_x）$和$\mathrm{LS}（\mu_x）$.我们证明了$\frac{1}{1+| X|}\le\martda_1（\mu_x）\le 1$而$C_{\martrm {LS}}（\mu_x）$的行为类似于$\log（1+\frac{1}{1-|X|}）$作为$|X| 1.$
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 $|x|\to 1.$