Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

Yutao Ma, Zhengliang Zhang
2014 Electronic Communications in Probability  
In this paper, we consider the circular Cauchy distribution µx on the unit circle S with index 0 ≤ |x| < 1 and we study the spectral gap and the optimal logarithmic Sobolev constant for µx, denoted respectively by λ1(µx) and CLS(µx). We prove that 1 1+|x| ≤ λ1(µx) ≤ 1 while CLS(µx) behaves like log(1 + 1 1−|x| ) as |x| → 1.
doi:10.1214/ecp.v19-3071 fatcat:xt6jymc27jalbbxt3fc3hbizxa