Finite-sample consistency of empirical e-KQD for p>1

Establish finite-sample consistency for the expected kernel quantile discrepancy e-KQD_p with integer p>1 by proving that, under the assumptions that X is a subset of R^d, the weighting measure ν on [0,1] has a density, the distributions P and Q on X have densities bounded away from zero, and E_{X∼P}[k(X,X)^{p/2}] and E_{X∼Q}[k(X,X)^{p/2}] are finite, the quantity E_{x_{1:n}∼P, y_{1:n}∼Q}[ | e-KQD_p(P_n, Q_n; ν, γ_l) − e-KQD_p(P, Q; ν, γ) | ] achieves the rate O(l^{-1/2} + n^{-1/2}).

Background

The paper introduces kernel quantile embeddings (KQEs) and kernel quantile discrepancies (KQDs), and establishes finite-sample consistency for the empirical e-KQD when p=1, achieving an O(l^{-1/2} + n^{-1/2}) rate under mild conditions (Theorem res:consistency_KQD).

For p>1, the straightforward reduction to empirical CDF convergence used for p=1 does not apply, and controlling higher-order moments involves integrability conditions expressed via J_p(R) and geometric properties of level sets through the coarea formula. Motivated by these considerations, the authors formulate a conjecture asserting that the same O(l^{-1/2} + n^{-1/2}) rate should hold for p>1 under additional assumptions on densities and moments, and they sketch a proof strategy but leave a rigorous proof open.