隐秘点过程与格诱导
Stealthy point processes and lattice induction
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中文总结 AI 辅助
研究基于格诱导证明隐秘点过程实现定理及逆定理，构造出\(\mathbb{R}^d\)上非纯点巴特利特谱的平移不变隐秘点过程，还给出遍历平移不变点过程是格诱导的条件，与相关定理作比较。
中文摘要 AI 辅助
我们基于格诱导证明了隐秘点过程的一个实现定理以及一维情况下的一个逆定理。从满秩格诱导出的每个保概率\(\mathbb{R}^d\)作用都允许一个生成的德洛内横截面，其巴特利特谱在原点的一个邻域上消失。可以选择构造使得一阶线性统计检测到诱导谱类型的非零部分。伯努利基产生非零的绝对连续分量，而具有奇异最大谱类型的弱混合基产生非零的奇异连续分量。据我们所知，这些是\(\mathbb{R}^d\)上具有非纯点巴特利特谱的首批严格构造的平移不变隐秘点过程。反之，设\(\eta\)是\(\mathbb{R}\)上具有正强度和局部二阶矩的遍历平移不变点过程。如果\(\int_{0<|\xi|<1}\frac{1}{\xi^2}\,d\sigma_\eta(\xi)<\infty\)，那么它的平移作用有一个非零特征值且是格诱导的。因此，一个遍历保概率博雷尔\(\mathbb{R}\)空间是格诱导的当且仅当它允许一个生成的隐秘德洛内横截面。这应与博里切夫、索丁和韦斯的一个定理相比较，该定理指出\(\mathbb{Z}\)上具有适当谱支撑的平移不变点过程是周期的。
英文摘要
We prove a realization theorem for stealthy point processes based on lattice induction, together with a converse in dimension one. Every probability-preserving $\mathbb R^d$-action induced from a full-rank lattice admits a generating Delone cross-section whose Bartlett spectrum vanishes on a neighborhood of the origin. The construction can be chosen so that first-order linear statistics detect a nonzero part of the inducing spectral type. Bernoulli bases yield a nonzero absolutely continuous component, while weakly mixing bases of singular maximal spectral type yield a nonzero singular-continuous component. To our knowledge, these are the first rigorously constructed translation-invariant stealthy point processes on $\mathbb R^d$ with non-pure-point Bartlett spectrum. Conversely, let $η$ be an ergodic translation-invariant point process on $\mathbb R$ with positive intensity and local second moments. If \[ \int_{0<|ξ|<1}\frac{1}{ξ^2}\,dσ_η(ξ)<\infty, \] then its translation action has a nonzero eigenvalue and is lattice-induced. Consequently, an ergodic probability-preserving Borel $\mathbb R$-space is lattice-induced if and only if it admits a generating stealthy Delone cross-section. This should be compared with a theorem of Borichev, Sodin and Weiss stating that a translation-invariant point process on $\mathbb Z$ with proper spectral support is periodic.
