Poincaré and log-Sobolev inequality for stationary Gaussian processes and moving average processes
Let \(X=(X_n)_{n\in\mathbb Z}\) be a real-valued stationary Gaussian process. The aim of the present paper is to characterize those processes \(X\) for which the Poincaré and/or the logarithmic Sobolev inequality hold in \(\mathbb R^{\mathbb Z}\). The characterization is in terms of the spectral measure \(\mu\) of \(X\). For example, \(X\) satisfies the above mentioned inequalities if and only if \(\mu\) is absolutely continuous with respect to the Lebesgue measure and, moreover, the density of \(\mu\) is bounded. The authors obtain also sharp bounds in these inequalities.
scientific article; zbMATH DE number 5016090