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Poincaré Inequality and Log-Sobolev Inequality for Stationary Gaussian Processes

Author: LiGuangFei
Tutor: WuLiMing
School: Wuhan University
Course: Probability and Statistics
Keywords: Poincare inequality log-Sobolev inequality Stationary Gaussian process Moving average process
CLC: O178
Type: Master's thesis
Year: 2005
Downloads: 47
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Abstract


This paper describes a stationary Gaussian process to meet the necessary and sufficient conditions are given Poincaré inequality and log-Sobolev inequality satisfy the corresponding inequalities optimal constant , and get the results to the moving average process . First of all, for the discrete-time stationary Gaussian process X : = ( X the n < / sub >) n ∈ Z set EX 0 2 = σ 2 gt; 0, EX 0 X n = σ 2 p (n), (? ) n ∈ Z. Let μ its nonnegative bounded spectral measure satisfying : the σ 2 < / sup> p (n ) = 1 / ( 2π) integral from n = -π to π e -int < / sup> dμ (t), (?) n ∈ Z, then X satisfy the Poincaré inequality or log-Sobolev inequality if and only if μ lt; lt; dt and the density f : = dμ / dt is bounded . The corresponding Poincaré inequality and log-Sobolev inequalities optimal constant is determined as follows: the ‖ of ‖ f ∞ = esssup t f (t ) . Second , given similar results for continuous-time stationary Gaussian process . Finally, the obtained results are extended to the moving average process , and two specific examples are given .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Inequality and other
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