I'm looking for references on the asymptotic distribution of V-statistics with 2nd order degenerate symmetric kernel $h:\mathcal{X}^k \to \mathbb{R}$ of degree $k\geq 2$. That is, we are considering the following V-statistic $$ V_n = \frac{1}{n^k}\sum_{i_1=1}^n\cdots \sum_{i_k=1}^n h(X_{i_1},...,X_{i_k}) $$ and the corresponding U-statistic $$ U_n = {n\choose k}^{-1}\sum_{1\leq i_1<\cdots <i_k \leq n} h(X_{i_1},...,X_{i_k}) $$ for some i.i.d. sequence $(X_n)_{n\in\mathbb{N}}$ of random elements in $\mathcal{X}$.

If $h$ is 2nd order degenerate we know that $ nU_n \Rightarrow \sum_{i=1}^\infty \lambda_i (Z_i^2-1), $ where $(Z_i)$ is an i.i.d. sequence of standard normal distributed random variables and $(\lambda_i)$ is the eigenvalues of a linear operator $S:L^2(\mathcal{X},P_X)\to L^2(\mathcal{X},P_X)$. The procedure used by Serfling(1980) to derive the asymptotic distribution of V-statistics from the asymptotic distribution of U-statistics in the non-degenerate case, can't be used when $h$ is degenerate.

However I have been some weak convergence results for V-statistics with degenerate kernels: Every monograph on U-statistics I have encountered ("Theory of U-statistics" (1994) by Borovskich and Koroljuk p. 141 & "Approximation Theorems of Mathematical Statistics" (1980) by Serfling p. 227) only shows the degenerate V-statistic weak convergence for $k=2$, that is if $h$ is a 2nd order degenerate kernel of degree 2, then (under some conditions ofcourse) $$ nV_n \Rightarrow \sum_{i=1}^\infty \lambda_i Z_i^2. $$

I need a similar result for a 2nd order degenerate kernel of degree $k>2$, can somebody provide me with a reference for such a result?



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