Completely Positive Entropy and Fourier Central Limit Theorems for Stationary Random Measures

Abstract

We prove an almost-everywhere Fourier central limit theorem for stationary random measures on $\bR^d$ with local second moments whose translation action is essentially free and has completely positive entropy. For the resulting almost-everywhere defined Bartlett density $s_{\eta }$, we show that there is a single ${\lambda }_d$-conull set of frequencies, independent of the test functions, on which finite collections of normalized smooth-window Fourier transforms converge jointly to proper complex Gaussian limits with covariance determined by $s_{\eta }$. No quantitative mixing, correlation-decay, or cumulant-summability assumption is imposed. For stationary point processes of positive intensity, the same good-frequency set yields Gaussian limits for ball-window Fourier transforms and exponential limits for their squared moduli. We also construct a stationary ergodic zero-entropy random measure with bounded continuous Bartlett density, positive ${\lambda }_1$-almost everywhere, for which the Fourier central limit theorem fails.

36 pages, comments are welcome!