Hermite ranks and $U$-statistics

Abstract

We focus on the asymptotic behavior of $U$-statistics of the type

$$\begin{aligned} \sum _{1\le i\ne j\le n} h(X_i,X_j)\\ \end{aligned}$$

in the long-range dependence setting, where $(X_i)_{i\ge 1}$ is a stationary mean-zero Gaussian process. Since $(X_i)_{i\ge 1}$ is Gaussian, $h$ can be decomposed in Hermite polynomials. The goal of this paper is to compare the different notions of Hermite rank and to provide conditions for the remainder term in the decomposition to be asymptotically negligeable.