The almost sure invariance principle for the empirical process of U- statistic structure
Consider an i.i.d. sequence \(X_ 1,X_ 2,..\). of real-valued random variables, and let \(g: {\mathbb{R}}^ m\to {\mathbb{R}}\) be a measurable kernel. If, for \(t\in {\mathbb{R}}\), \[ \mu_ n(t)=\left( \begin{matrix} n\\ m\end{matrix} \right)^{-1}card\{1\leq i_ 1<...<i_ m\leq n: g(X_{i_ 1},...,X_{i_ m})\leq t\} \] and \(\mu (t)=E\mu_ n(t)\), then \(\{n^{1/2}(\mu_ n(t)-\mu (t))\), \(n\geq 1\}\) defines the empirical process of U-statistic structure. The authors' main result is provided by a strong approximation of \(n(\mu_ n-E\mu_ n)\) to a certain Gaussian process up to an order \(O(n^{1/2-\lambda})\), \(\lambda >0\). Some applications are also discussed, e.g. a functional law of iterated logarithm for \(\mu_ n\) or a law of iterated logarithm for generalized L-statistics.
empirical process of U-statistic structure
law of iterated logarithm for generalized L-statistics
scientific article; zbMATH DE number 4005248