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A Φ-Entropy Contraction Inequality for Gaussian Vectors

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Abstract

A beautiful result of Sarmanov (Dokl. Akad. Nauk SSSR 121(1), 52–55, 1958) says that for a Gaussian vector (X,Y), \(\operatorname {Var}(\mathbb {E}[f(Y)|X])\le \rho^{2}\operatorname {Var}(f(Y))\) for all measurable functions f, where ρ is the (linear) correlation coefficient between X and Y. We generalize this result to a general Φ-entropy (a nonlinear version of his result) by means of a previous result of D. Chafai based on Bakry–Emery’s Γ 2-technique and tensorization.

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References

  1. Bakry, D., Emery, M.: Diffusions hypercontractives. In: Séminaire de Probabilités. Lect. Notes Math., vol. 1123, pp. 259–206. Springer, Berlin (1985)

    Google Scholar 

  2. Chafai, D.: Entropies, convexity, and functional inequalities: on Φ-entropies and Φ-Sobolev inequalities. J. Math. Kyoto Univ. 44(2), 325–363 (2004)

    MathSciNet  Google Scholar 

  3. Chafai, D.: Binomial-Poisson inequalities and M/M/∞ queue. ESAIMS Probab. Stat. 10, 317–339 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Del Moral, P., Ledoux, M., Miclo, L.: On contraction properties of Markov kernels. Probab. Theory Relat. Fields 126(3), 395–420 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  5. Kolmogorov, A.N., Rozanov, Yu.A.: On strong mixing conditions for stationary Gaussian processes. Theory Probab. Appl. 5(2), 204–208 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  6. Latala, R., Oleszkiewicz, K.: Between Sobolev and Poincaré. In: Geometric Aspects of Functional Analysis. Lect. Notes Math., vol. 1745, pp. 147–168. Springer, Berlin (2000)

    Chapter  Google Scholar 

  7. Massart, P.: Concentration inequalities and model selection. In: Saint-Flour Summer School, 2003

  8. Sarmanov, O.V.: Maximal coefficient of correlation (nonsymmetric case). Dokl. Akad. Nauk SSSR 121(1), 52–55 (1958) (in Russian)

    MATH  MathSciNet  Google Scholar 

  9. Wu, L.: Large and moderate deviations and exponential convergence for stochastic damping Hamilton systems. Stoch. Process. Appl. 91, 205–238 (2001)

    Article  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, CNRS-UMR 6620, Université Blaise Pascal, 63177, Aubière, France

    Liming Wu

  2. Department of Mathematics, Wuhan University, 430072, Wuhan, China

    Liming Wu

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  1. Liming Wu
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Correspondence to Liming Wu.

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Supported by the ANR Project “Inégalités fonctionelles” and the NSF of China.

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Wu, L. A Φ-Entropy Contraction Inequality for Gaussian Vectors. J Theor Probab 22, 983–991 (2009). https://doi.org/10.1007/s10959-009-0211-0

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  • DOI: https://doi.org/10.1007/s10959-009-0211-0

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