Optimal Bounds in Non-Gaussian Limit Theorems for U-Statistics

Abstract

Let $X,X_1,X_2,\ldots$ be i.i.d. random variables taking values in a measurable space $\mathscr{X}$. Let $\phi(x,y)$ and $\phi_1(x)$ denote measurable functions of the arguments $x,y\in\mathscr{X}$. Assuming that the kernel $\phi$ is symmetric and the $\mathbf{E}\phi(x,X)= 0$, for all $x$, and $\mathbf{E}\phi_1(X) = 0$, we consider $U$-statistics of type $$T = N^{-1} \textstyle\sum\limits_{1\le j < k\le N} \phi(X_j,X_k) + N^{-1/2} \textstyle\sum\limits_{1\le j\le N}\phi_1(X_j). $$ Is is known that the conditions $\mathbf{E}\phi^2(X,X_1)<\infty$ and $\mathbf{E}\phi_1^2(X)<\infty$ imply that the distribution function of $T$, say $F$, has a limit, say $F_0$, which can be described in terms of the eigenvalues of the Hilbert-Schmidt operator associated with the kernel $\phi(x,y)$. Under optimal moment conditions, we prove that $$\Delta_N = \sup_x |F(x) - F_0(x) - F_1(x)|= \mathcal{O}(N^{-1}), $$ provided that at least nine eigenvalues of the operator do not vanish. Here $F_1$ denotes an Edgeworth-type correction. We provide explicit bounds for $\Delta_N$ and for the concentration functions of statistics of type $T$.