AI 中文总结
本文针对$\bR^d$上满足局部二阶矩、平移作用本质自由且具完全正熵的平稳随机测度，证明了傅里叶中心极限定理，还构造了反例说明零熵情形下该定理不成立。
AI 中文摘要
我们证明了：对于$\bR^d$上具有局部二阶矩、平移作用本质自由且具有完全正熵的平稳随机测度，存在几乎处处成立的傅里叶中心极限定理。对由此得到的几乎处处定义的Bartlett密度$s_\boldsymbol{\u03b7}$，我们证明存在一个单一的$\boldsymbol{\u03bb}_d$-零测度频率集（与检验函数无关），在该集上，有限组归一化光滑窗傅里叶变换会联合收敛到恰当的复高斯极限，其协方差由$s_\boldsymbol{\u03b7}$确定。本文未施加任何定量混合、相关衰减或累积量可和性假设。对于正强度的平稳点过程，同一良好频率集会为球窗傅里叶变换给出高斯极限，并为其平方模给出指数极限。我们还构造了一个具有有界连续Bartlett密度、在$\boldsymbol{\u03bb}_1$-几乎处处为正的平稳遍历零熵随机测度，对其而言傅里叶中心极限定理不成立。
英文摘要
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 $s_η$, we show that there is a single $λ_d$-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 $s_η$. 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 $λ_1$-almost everywhere, for which the Fourier central limit theorem fails.
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