Some moment and exponential inequalities for \(V\)-statistics with bounded kernels (Q5939313)

This article provides moment and exponential inequalities for degenerate \(V\)-statistics with bounded kernels as an application of inequalities for sub-Bernoulli functions (functions of several independent variables whose conditional moments are dominated by those of products of independent Bernoulli variables). In particular sharp multiplicative constants are obtained for the term in \(t^{2/k}\), in the exponent, where \(k\) is the order of the \(V\) (or \(U)\) statistic. On the other hand, these inequalities do not detect the Gaussianness of the tail probabilities of the degenerate \(V\) or \(U\) statistic for relatively small values of \(t\) [for \(k=2\), compare with the exponential inequality on page 36 of \textit{E. Giné}, \textit{R. Latała} and \textit{J. Zinn}, in: High dimensional probability II. Prog. Probab. 47, 13-38 (2000; Zbl 0969.60024)].