A Martingale Kernel Independence Test

The Hilbert-Schmidt Independence Criterion (HSIC) and its joint-independence extension dHSIC are degenerate V-statistics whose data-dependent weighted-chi2 null limits force a permutation calibration that multiplies the per-test cost by the number of permutations, in practice two orders of magnitude. Adapting the recent martingale MMD construction for two-sample testing to the (joint) independence problem, we introduce two studentised statistics whose null distributions are standard normal regardless of the data law, so that a single normal-quantile lookup replaces the permutation step entirely. The first, mHSIC, is a self-normalised lower-triangular sum of the Hadamard product of two empirically centred Gram matrices. Under independence and bounded-fourth-moment kernels it converges to a standard normal. It is consistent against every fixed alternative, and runs at quadratic cost in the sample size without any sample split, matching the biased HSIC V-statistic. Our second statistic, mdHSIC, achieves finite-sample consistency with a single half-sample split: the centring is estimated on one half and the lower-triangular self-normalised martingale is run on the other, shrinking the conditional-mean residual to a quantity that is exponentially small in d, so the statistic is asymptotically standard normal at every fixed number of jointly tested variables, with a per-test cost that grows only linearly in d. On synthetic data with per-variable input dimension from 1 to 500 and between 2 and 10 jointly tested variables, both statistics match the empirical type-I error rate and test power of permutation-calibrated baselines while running 25 to 60× faster.