Randomized Quadrature with Periodic Kernels: Applications to Cavalieri Volume Estimation

Abstract

This paper studies randomized algorithms for unbiased numerical integration of d-dimensional periodic functions using kernel-based quadrature rules, with particular emphasis on rules induced by periodic radial basis function (RBF) kernels. The integration points are either deterministically generated or locally perturbed and then randomly shifted, introducing structured randomness into the scheme. The analysis builds on tools from the theory of reproducing kernel Hilbert spaces (RKHS) and Sobolev interpolation. It is shown that the resulting estimators achieve optimal variance decay rates, effectively capturing the smoothness of the integrand even when the assumed regularity is overestimated. The work is motivated by Cavalieri volume estimation, a classical problem in stereology. The theoretical results generalize this framework to higher dimensions and provide a Fourier-based perspective on smoothness, yielding a flexible and mathematically grounded alternative for randomized quadrature with periodic structure.