The central limit theorem for \(U\)-processes indexed by Hölder's functions
Given a probability \(P\) on \(R\), a class of real functions \(\mathcal F\) and an i.i.d. sequence of r.v.s \(\{X_ i\}\) with law \(P\), \(\mathcal F\) is said to be \(P\)-Donsker if \(\{n^{1/2} \sum^ n_ 1 (f(X_ i - Pf))\}_{f \in {\mathcal F}}\) converges weakly to a Radon limit as a random element of \(\ell^ \infty({\mathcal F})\). \textit{E. Giné} and \textit{J. Zinn} [Ann. Probab. 14, 1329-1338 (1986; Zbl 0611.60029)] showed that \({\mathcal F}_ 1\), the class of Hölder's functions of order one, is \(P\)-Donsker iff \(\sum^ \infty_{k = 1} (P(k - 1 < | x | \leq k))^{1/2} < \infty\). The author extends this result for \(1/2 < \alpha \leq 1\) and a positive integer \(m\), to \(\{n^{1/2} (U^ m_ n (f) - Ef\}_{{\mathcal F}_ \alpha}\), which is a generalization of empirical processes, known as \(U\)-processes. For canonical \(U\)-processes there is no similar result. In fact, the author proves that if \(P\) is equal to the Lebesgue measure on \([0,1]\) and \(m \geq 2\), \(\{n^{m/2} U^ m_ n (\pi_{m,m} f) : f \in {\mathcal F}_ 1 ([0,1]^ m)\}\) does not converge to a Radon limit.
scientific article; zbMATH DE number 633664