A remark on convergence in distribution of \(U\)-statistics
For a sequence of i.i.d. r.v.'s \(X_ j\) with values in a measurable space \((S, {\mathcal S}, P)\) and a measurable symmetric function \(h:S^ m \to \mathbb{R}\) one considers \(U\)-statistics \[ U_ n^{(m)} (h) = {n \choose m}^{-1} \sum_{i_ 1 < \cdots < i_ m \leq n} h(X_{i_ 1}, \dots, X_{i_ m}). \] It is proved that if the sequence \(\{n^{m/2} U_ n^{(m)} (h)\}\) is stochastically bounded, then \(Eh^ 2(X_ 1, \dots, X_ m)\) \(< \infty\) and \(Eh(X_ 1, x_ 2, \dots, x_ m) = 0\) for a.e. \((x_ 2, \dots, x_ m) \in S^{m-1}\) (hence \(\{n^{m/2} U_ n^{(m)} (h)\}^ \infty_{n = 1}\) converges in distribution). This provides for the case \(m=1\) a new elegant proof of the classical result by Feller, Khinchin, Lévy on the necessity of \(EX^ 2_ 1 < \infty\) for CLT. An analogue of the main result is given also for \(h:S^ m \to B\), \(B\) being a Banach space of cotype 2 (if \(B\) is not of cotype 2, then the assertion is not true even for \(m=1)\).
scientific article; zbMATH DE number 567176