Poincaré Inequality And Log-Sobolev Inequality For Stationary Gaussian Processes

For stationary Gaussian processes, we obtain the necessary and sufficient conditions for Poincare inequality and log-Sobolev inequality of process-level and provide the sharp constants, then we extend the results to moving average processes. Firstly, let be a real valued stationary Gaussian process with .Let μ be the nonnegative and bounded spectral measure on the torus T identified as [-π,π], which is determined bythen X satisfies the Poincare or log-Sobolev inequality iff μ << dt and the density f := dμ/dt is bounded. The according constants to the equalities are:where ||f||_∞ = esssup_t/(t). Secondly, it is similar for the continuous time counterpart. Finally, the extension to moving average processes is also presented, as well as several concrete examples.