Abstract:We introduce a general-dimensional origin-invariant Cramér--von Mises statistic for testing complete spatial randomness (CSR), defined as the average of corner-oriented Cramér--von Mises functionals over all \(2^d\) corners of the unit cube. The statistic admits a closed-form \(O(n^2)\) computing formula and is therefore directly implementable in arbitrary dimension. In dimension one, it reduces to the classical rank-based Cramér--von Mises statistic; in dimension two, it agrees with Zimmerman's origin-invariant statistic. Under the CSR null hypothesis, the finite-dimensional distributions of the normalized empirical process converge jointly to a mean-zero Gaussian process with an explicit cross-corner covariance kernel. We establish the existence of the limiting process, construct a continuous modification, and prove asymptotic equicontinuity using Bernstein chaining. The resulting quadratic limit is characterized through the covariance operator on the corner--location product space and has a Karhunen--Loève representation. An ordered-list construction and restriction law connect the conditional CSR model with the homogeneous Poisson point-process formulation. Monte Carlo null percentiles are reported for dimensions one through five and agree with the exact identity \(\mathbb{E}[n\bar{\omega}_d^2]=2^{-d}-3^{-d}\). The principal foundational results are formally verified in Lean 4 and mathlib4.
From: Marco Mandap PhD [view email]
[v1]
Mon, 18 May 2026 22:32:52 UTC (31 KB)
[v2]
Wed, 22 Jul 2026 01:34:00 UTC (33 KB)
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