Limit Theorems for $U$-Processes

Exponential and Moment Inequalities for U-Statistics

“…The moment inequalities in the first part of this article do imply exponential bounds for canonical U -statistics of any order and with bounded kernels which are sharper than those in Arcones and Giné (1993); however, they are not of the best kind as they do not exhibit Gaussian behavior for part of the tail, which they should in view of the tail behavior of Gaussian chaos.…”

“…Inequality (3.5) is better than the Bernstein type inequality in Arcones and Giné (1993) as it is better for x ≤ n 2 A and the probability is zero for x ≥ n 2 A. Inequality (3.5) is suboptimal for small values of x, for which the exponent should be a constant times −x 2 , just as for chaos variables of order 2 (see Ledoux and Talagrand (1991) and Lata la (1999)). This suggest that inequality (2.9) is not of the best kind, and can be improved.…”

Stability of precise Laplace's method under approximations; Applications

“…1 and Cor. 1) (see also Arcones-Gine [1], Th. 2.1, or the appendix), there exists universal constants C 0 and C 0 so that the following randomization inequality holds…”

“…It is not a priori clear since the error one should make by a trivial bound should be of order e o(N ) . Its proof is driven by the control on U -statistics developed by De La Peña [15] and Arcones-Gine [1]. The article is organized as follows.…”

“…These estimates are based on the works of De La Peña [15] and Arcones-Gine [1]. We first study the two-body setting and denote…”