Martingalized U-statistics-based sequential inference: a blessing of degeneracy

Degenerate degree-two U-statistics have non-Gaussian limits, making sequential inference substantially more difficult than in the nondegenerate case. However, row normalization preserves the target parameter and makes the centered numerator a martingale on a harmonic variance clock, at the cost of logarithmic variance inflation. We establish a strong Gaussian approximation for this process and consistently estimate its variance through online updates, allowing spectrum-free calibration. Combining this strong approximation with Gaussian-mixture (GM) and stitched law-of-the-iterated-logarithm (LIL) boundaries yields confidence sequences with asymptotic time-uniform coverage and corresponding sequential tests for null hypotheses under which the kernel is degenerate. We further extend the framework to multi-dimensional vector-valued kernels, yielding Mahalanobis confidence ellipsoids based on consistent covariance estimation. Numerical experiments illustrate sequential type-I error control and favorable detection power in two-sample and independence testing against spectral baselines, and in three-sample homogeneity testing using vector confidence ellipsoids.