A note on the rate of convergence of the bootstrap (Q1308793)

Suppose \((X_ i)_{i\in\mathbb{N}}\) is an i.i.d. sequence of random variables defined on some probability space \((\Omega,{\mathcal A},\mathbb{P})\) with distribution function (d.f.) \(F\), \(F_{\eta_ n}\) denotes the d.f. of the standardized sample mean \(\eta_ n=\Bigl(\sum^ n_{i=1} X_ i- \mu_ n\Bigr)/(\sigma n^{1/2})\), \(\mu=\mathbb{E}(X_ 1)\) the expectation of \(X_ 1\), \(\sigma^ 2=\text{var}\) \((X_ 1)\) the variance, and \(F_ 0\) the standard named d.f. If the third moment of \(X_ 1\) exists, \(\mathbb{E}(| X_ 1|^ 3)<\infty\). \textit{P. L. Butzer}, \textit{L. Hahn} and \textit{U. Westphal} [J. Approximation Theory 13, 327-340 (1975; Zbl 0298.60014)], \[ \sup_{y\in\mathbb{R}}\Bigl| \int f(x+ y)F_{\eta_ n}(dx)-\int f(x+y)F_ 0(dx)\Bigr|=O(n^{-1/2}),\quad n\to\infty \] for all \(f\in C^ 3_ p=\{g\in C^ 0_ b: g^{(i)}\in C^ 0_ b,\;1\leq i\leq 3\}\), where \(g^{(i)}\) denotes the \(i\)th derivative of \(g\), and \(C^ 0_ b\) the class of bounded and uniformly continuous functions defined on \(\mathbb{R}\). The author proves that the ``large O'' approximation can also be improved to a ``small o'' result if the bootstrap d.f. \(F_{\eta^*_ n}\) is used instead of the normal d.f. \(F_ 0\) (\(F_{\eta^*_ n}\) denotes the d.f. of the standardized sample mean of the bootstrap sample \((X^*_ i)_{i=1,\dots,n}\) which is an i.i.d. sample according to the empirical d.f. \(F_ n\) of the \(X\)-sample), i.e. the following is true: Theorem. Assume that \(\mathbb{E}(| X_ 1|^ 3)<\infty\). Then, with probability one, \[ \sup_{y\in\mathbb{R}}\Bigl| \int f(x+ y)F_{\eta_ n}(dx)- \int f(x+y) F_{\eta^*_ n}(dx)\Bigr|=o(n^{-1/2}),\quad n\to\infty, \] for all \(f\in C^ 3_ b\).