Bracketing metric entropy rates and empirical central limit theorems for function classes of Besov- and Sobolev-type (Q2641420)

The authors derive upper bounds for the \({\mathcal L}^r(\mu)\)-bracketing metric entropy and sup-norm metric entropy of bounded subsets \(\mathcal F\) of some (weighted) function spaces defined on \({\mathbb R}^n\). Real Besov spaces \(B^s_{p,q}({\mathbb R}^n, \langle x\rangle^\beta)\), real Sobolev spaces \(H^s_{p}({\mathbb R}^n, \langle x\rangle^\beta)\) with polynomial weights \(\langle x\rangle^\beta=(1+| x|^2)^{\beta/2}\) are regarded, as well as Hölder spaces \({\mathcal C}^s({\mathbb R}^n)\). It is assumed that \(s-\frac n p >0\); so the spaces consist of regular distributions. Moreover, upper bounds for the metric entropies of the restrictions of sets \(\mathcal F\) to Borel subsets \(\Omega\subset{\mathbb R}^n\) are calculated. The results for Besov spaces imply the similar estimates for Triebel--Lizorkin spaces. The estimates are applied to the theory of empirical processes. In particular, it is proved that the above bounded subsets are Donsker classes uniformly in various sets of probability measures. To prove the main estimates the authors adapt results of \textit{D.\,Haroske} and \textit{H.\,Triebel} [Math.\ Nachr.\ 167, 131--156 (1994; Zbl 0829.46019); ibid.\ 278, 108--132 (2005; Zbl 1078.46022)].