Asymptotic distributions of degenerated U-statistics

A degenerated U-statistic with rank 2 is known to converge in distribution to a weighted sum of independent chi-square random variables. However, a result for the asymptotic distribution of a degenerated U-statistic with rank higher than 2 is not available in the literature. We derive explicitly the asymptotic distribution of a degenerated U-statistic with any rank, which is useful for hypothesis testing when the test statistic is a degenerated U-statistic under null hypothesis. Interestingly, the limit for a degenerated U-statistic with rank higher than 2 is a weighted sum of independent polynomials of the standard normal random variables.