Rates of Convergence for Some Functionals in Probability

摘要: Let $\{x_1, x_2,\cdots\}$ be a sequence of i.i.d.r.v. with mean zero, variance one, and (1) $\mathbf{P}(|x_k| \geqq \lambda) \leqq C \exp(-\alpha\lambda^\varepsilon)$ for positive $\alpha, \varepsilon$. $f(t, x)$ (with its first partial derivatives) slow growth in $x$, let $F_n(x)$ the distribution function $(1/n) \sum^n_1 f(k/n, s_k/n^{\frac{1}{2}})$ where $s_k = x_1 + x_2 \cdots x_k$, $F(x)$ $\int^1_0 f(t, w(t)) dt$ $\{w(t)\}$ is Brownian motion. Then $\sup_x |F_n(x) - F(x)| O((\log n)^\beta/n^{\frac{1}{2}})$ provided has bounded derivative. The proof uses Skorokhod representation; also, theorem proven which would indicate that representation cannot used general to obtain rate convergence better than $O(1/n^{\frac{1}{4}})$. A corresponding result obtained if replaced by existence finite $p$th moment, $p 4$.