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Brownian sheet and uniformity tests on the hypercube
A. Cabaña, E. M. Cabaña · 2025-09-08 · via math.ST updates on arXiv.org

A construction of $p$-parameter Brownian sheet on the hypercube $C=[0,1]^p$ as a sum of $2^p$ independent Gaussian processes is obtained. The terms are closely related to Brownian pillows, and the probability laws of their $L^2(C)$ squared norms are computed. This allows us to propose consistent tests of uniformity for samples of i.i.d. random vectors on $C$. A comparison of powers of the new tests with those of several uniformity tests found in the statistical literature completes the article. The proposed tests show a good performance in detecting copula alternatives. Keywords: Brownian sheet, multivariate uniformity tests.

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