Large deviations of U-statistics. I (Q1881784)

Large deviation results are proved for non-degenerate \(U\)-statistics of degree \(m\) of the form \[ U_n={(m-1)\cdots 2\cdot 1 \over{(n-1)\cdots (n-m+1) } } \sum_{1\leq i_1 < \cdots < i_m \leq n } h(X_{i_1}, \ldots, X_{i_m}), \] where \(X_1,\ldots X_n\) be independent and identically distributed random variables, taking values in a measurable space, the kernel \(h\) is an integrable symmetric function. Under Cramér's condition \( E\, \exp \{a| h(X_1,\ldots, X_m)| \}<\infty \) for some \(a>0\), the large deviations equality is proved for \(x=O(\sqrt{n}/ \ln_l n)\). Here \(\ln_l n=\ln \ln_{l-1}n \), \(\ln_0 n = n\), \(l\geq 1\). An analogous result (Theorem 3) is proved for \(x=o(n^{\alpha})\) under Linnik's condition \( E\, \exp \{a_{\alpha}| h| ^{4\alpha/(2\alpha +1)} \}<\infty\), \(0<\alpha <1/2, \) for some \(a_{\alpha}>0\). The method of proof is based on the contraction technique of \textit{R. W. Keener, J. Robinson} and \textit{N. C. Weber} [Stat. Probab. Lett. 37, No. 1, 59--65 (1998; Zbl 0928.60016)], which is a natural generalization of the classical Cramér's method. [For part II see below, Zbl 1062.60020.]