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

Bentkus, V. ; Götze, F.

Ann. Probab., Tome 27 (1999) no. 1, p. 454-521 / Harvested from Project Euclid

Résumé

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

Détail

Publié le : 1999-01-14
Classification:  $U$-statistics,  degenerate $U$-statistics,  von Mises statistics,  symmetric statistics,  central limit theorem,  convergence rates,  Berry-Esseen bounds,  Edgeworth expansions,  second order efficiency,  62E20,  60F05