Logarithmic Sobolev and Poincaré inequalities for the circular Cauchy distribution

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

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 $|x|\to 1.$

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-9

Publication Date: February 18, 2014

DOI: 10.1214/ECP.v19-3071

References

  • Bakry, D.; Émery, Michel. Diffusions hypercontractives. (French) [Hypercontractive diffusions] Séminaire de probabilités, XIX, 1983/84, 177--206, Lecture Notes in Math., 1123, Springer, Berlin, 1985. MR0889476
  • Barthe, F.; Roberto, C. Sobolev inequalities for probability measures on the real line. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). Studia Math. 159 (2003), no. 3, 481--497. MR2052235
  • Barthe, F., Ma, Y-T. and Zhang, Z.: Logarithmic Sobolev inequalities for harmonic measures on spheres. phJ. Math. Pures Appl., (2013) http://dx.doi.org/10.1016/j.matpur.2013.11.008.
  • Cattiaux, Patrick; Guillin, Arnaud. On quadratic transportation cost inequalities. J. Math. Pures Appl. (9) 86 (2006), no. 4, 341--361. MR2257848
  • Chen, Mufa. Analytic proof of dual variational formula for the first eigenvalue in dimension one. Sci. China Ser. A 42 (1999), no. 8, 805--815. MR1738551
  • Chen, Mufa; Zhang, Yuhui; Zhao, Xiaoliang. Dual variational formulas for the first Dirichlet eigenvalue on half-line. Sci. China Ser. A 46 (2003), no. 6, 847--861. MR2029196
  • Durrett, Richard. Brownian motion and martingales in analysis. Wadsworth Mathematics Series. Wadsworth International Group, Belmont, CA, 1984. xi+328 pp. ISBN: 0-534-03065-3 MR0750829
  • Emery, M. and Yukich, J.: A simple proof of logarithmic Sobolev inequality on the circle. phSéminaire de probabilités, 97, (1975), 1061-1083.
  • Kakutani, Shizuo. On Brownian motions in $n$-space. Proc. Imp. Acad. Tokyo 20, (1944). 648--652. MR0014646
  • Kato, Shogo. A Markov process for circular data. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 (2010), no. 5, 655--672. MR2758240
  • McCullagh, Peter. Möbius transformation and Cauchy parameter estimation. Ann. Statist. 24 (1996), no. 2, 787--808. MR1394988
  • Schechtman, G. and Schmuckenschläger, M.: A concentration inequality for harmonic measures on the sphere, phGeometric aspects of funct. analysis (Israel, 1992-1994): 255-273, phOper. Theory Adv. Appl., 77, (1995), Birkhäuser, Basel, 60-65 (31B99). MR1353465
  • Talagrand, M. Transportation cost for Gaussian and other product measures. Geom. Funct. Anal. 6 (1996), no. 3, 587--600. MR1392331
  • Zhang, Z.L., Ma, Y-T. and Lei, L.: Logarithmic Sobolev inequalities for Moebius measures on spheres. Submitted, 2013.
  • Zhang, Zhengliang; Miao, Yu. An equivalent condition between Poincaré inequality and $T_2$-transportation cost inequality. Acta Appl. Math. 110 (2010), no. 1, 39--46. MR2601641


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.