A class of asymptotically normal degenerate quasi U -statistics

Some quasi U -statistics, unlike other variants of U -statistics, arising in distance based tests for homogeneity of groups, have first-order stationary kernels of degree 2, and yet they enjoy asymptotic normality under suitable hypotheses of invariance. Central limit theorems for a more general class of quasi U -statistics with possibly higher order stationarity (and degree) are formulated with the aid of appropriate martingale (array) characterizations as well as permutational invariance structures.