The Annals of Probability

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

V. Bentkus and F. Götze

Full-text: Open access

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

\[ T = N^{-1} \textstyle\sum\limits_{1\le j < k\le N} \phi(X_j,X_k) + N^{-1/2} \textstyle\sum\limits_{1\le j\le N}\phi_1(X_j). \]

Abstract

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

Abstract

\[ \Delta_N = \sup_x |F(x) - F_0(x) - F_1(x)|= \mathcal{O}(N^{-1}), \]

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 $T$.

Article information

Source
Ann. Probab., Volume 27, Number 1 (1999), 454-521.

Dates
First available in Project Euclid: 29 May 2002

Permanent link to this document
https://projecteuclid.org/euclid.aop/1022677269

Digital Object Identifier
doi:10.1214/aop/1022677269

Mathematical Reviews number (MathSciNet)
MR1681161

Zentralblatt MATH identifier
1008.62017

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

Keywords
$U$-statistics degenerate $U$-statistics von Mises statistics symmetric statistics central limit theorem convergence rates Berry-Esseen bounds Edgeworth expansions second order efficiency

Citation

Bentkus, V.; Götze, F. Optimal Bounds in Non-Gaussian Limit Theorems for U -Statistics. Ann. Probab. 27 (1999), no. 1, 454--521. doi:10.1214/aop/1022677269. https://projecteuclid.org/euclid.aop/1022677269


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