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A Φ-Entropy Contraction Inequality for Gaussian Vectors

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  • Liming Wu

    (Université Blaise Pascal
    Wuhan University)

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.

Suggested Citation

  • Liming Wu, 2009. "A Φ-Entropy Contraction Inequality for Gaussian Vectors," Journal of Theoretical Probability, Springer, vol. 22(4), pages 983-991, December.
    • Handle: RePEc:spr:jotpro:v:22:y:2009:i:4:d:10.1007_s10959-009-0211-0
    DOI: 10.1007/s10959-009-0211-0
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    References listed on IDEAS

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    1. Wu, Liming, 2001. "Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems," Stochastic Processes and their Applications, Elsevier, vol. 91(2), pages 205-238, February.
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    Keywords

    Gaussian vectors; Φ-entropy;

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