Bracketing smooth functions
Let \({\mathcal F}\) be the class of functions \(f: R^ d\to R\) which are \(\alpha\)-times differentiable with derivatives bounded by numbers \(M_ j\) on each given set \(I_ j\) in a partition of \(R^ d= \bigcup_ j I_ j\). Upper and lower bounds on the \(L_ 2(P)\)-bracketing entropy of \({\mathcal F}\) are obtained. Some consequences of the bounds for empirical laws of large numbers and central limit theorems are given.
scientific article; zbMATH DE number 672062