2024/11/13 by Manjunath Krishnapur, Krishnapur, Manjunath, D. Yogeshwaran +1 · 2 citations
Decision Sciences · #60D05 #60F05 #60G55 #60G57 #FOS: Mathematics #Probability (math.PR) #Probability and Risk Models
paper · pdf · doi:10.48550/arxiv.2411.08848
openalex publication_date 2024/11/13 · openalex created_date 2024/11/16 · openalex updated_date 2026/07/28
We consider covariance asymptotics for linear statistics of general stationary random measures in terms of their truncated pair correlation measure. We give exact infinite series-expansion formulas for covariance of smooth statistics of random measures involving higher-order integrals of the truncated correlation measures and higher-order derivatives of the test functions and also equivalently in terms of their Fourier transforms. Exploiting this, we describe possible covariance and variance asymptotics for Sobolev and indicator statistics. In the smooth case, we show that that order of variance asymptotics drops by even powers and give a simple example of random measure exhibiting such a variance reduction. In the case of indicator statistics of C1-smooth sets, we derive covariance asymptotics at surface-order scale with the limiting constant depending on intersection of the boundaries of the two sets. We complement this by a lower bound for random measures with a non-trivial atomic part. Restricting to simple point processes, we prove a central limit theorem for Sobolev and Holder continuous statistics of simple point processes satifying a certain integral identity for higher-order truncated correlation functions.