Large deviations for \(U\)-statistics (Q5906522)

Let \(X_ 1, X_ 2, \dots\) be a sequence of independent random variables having common probability distribution \(F\), and let \(\varphi (x_ 1, \dots, x_ m)\) be a function (kernel) assumed (without loss of generality) to be symmetric in its arguments. For each such kernel we consider the associated \(U\)-statistic \[ U_ n={n \choose m}^{-1} \sum_{C_{nm}} \varphi (X_{i_ 1}, \dots, X_{i_ m}), \tag{1} \] where \(C_{nm}\) denotes the set of \(m\)-tuples \(\{(i_ 1, \dots, i_ m)\): \(1 \leq i_ 1<\cdots<i_ m \leq n\}\). Assume \[ E \varphi (X_ 1, \dots, X_ m) = 0 \tag{2} \] and suppose that \(\psi_ 1(X_ 1)=E (\varphi (X_ 1, \dots, X_ m) \mid X_ 1)\) has a positive variance \(\sigma^ 2\), i.e., \[ \sigma^ 2=E \psi^ 2_ 1 (X_ 1)>0. \tag{3} \] The purpose of this paper is to study the behavior of large-deviation probabilities for \(U\)-statistics.