The Spatial Cram'{e}r--von Mises Test of Independence under $\beta$-Mixing: Asymptotic Theory and Python Implementation

We derive the asymptotic distribution of the spatial Cram'{e}r--von Mises statistic for testing bivariate independence in stationary random fields on $\mathbb{R}^2$ under polynomial $\beta$-mixing dependence, and document the Python implementation that reproduces all simulation results. The classical test assumes i.i.d. observations; we extend it to spatially dependent data by combining three ingredients: (i) a Davydov-type covariance bound yielding integrability of the spatial covariance kernel under $\theta > 2(2+\delta)/\delta$; (ii) a reformulation of the inner-form test statistic as a degenerate U-statistic of order~2 with product kernel $Q = G_1 \otimes G_2$, following De Wet (1980); and (iii) an extension of Gregory's (1977) U-statistic limit theorem to $\beta$-mixing sequences via Yoshihara (1976). The limit distribution is a weighted sum of correlated $\chi^2_1$ variables whose eigenvalues factor as products of marginal eigenvalues; in the small-bandwidth limit the correlation vanishes and the limit reduces to the classical i.i.d. form. Explicit eigenvalue formulas are given for three weight functions (uniform, optimal normal, Anderson--Darling), producing computable critical values. The software generates Mat'{e}rn random fields by circulant embedding, computes the test statistic via the inner-form kernel decomposition, evaluates asymptotic critical values by Monte Carlo, and runs permutation-based alternatives. Simulation experiments show that the Anderson--Darling weight achieves the best power, while the Mantel and cross-$K$ tests have no power against cross-dependence in spatially correlated fields.