Let $\mathcal{M}$ denote the class of randomised monotone functions on $\mathbb{R}$ with values in $[0,1]$, and let $U_{\mathcal{M}}\colon \mathbb{R}_+\to \mathbb{R}_+$ be the minimal function for which \[ \mathbb{P}\left\{ \sqrt{η_f}\, \sup_{t\in\mathbb{R}} \left| f_Z(t) - \Exf{f_Z(t)} \right| \ge \varepsilon\sqrt{U_{\mathcal{M}}(η_f)} \right\} \le 2\mathrm{e}^{-2\varepsilon^2} \] holds for every member $f_Z$ of $\mathcal{M}$ of finite effective sample size $η_f$ and every positive $\varepsilon$. We prove that for every $x> 1$, \[ \left| \sqrt{U_{\mathcal{M}}(x)} - \sqrt{\log_4 x} \right| \le 2 \min\!\left\{ 1,\, \frac{2 \ln(\mathrm{e} + \ln x)}{\sqrt{\ln x}} \right\}\,. \] The optimal scale $\sqrt{U_{\mathcal{M}}(x)}$ is sharply tied, uniformly at finite sample sizes, to $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$.

翻译：设 $\mathcal{M}$ 表示 $\mathbb{R}$ 上取值于 $[0,1]$ 的随机单调函数类，且 $U_{\mathcal{M}}\colon \mathbb{R}_+\to \mathbb{R}_+$ 为使对 $\mathcal{M}$ 中每个具有有限有效样本量 $η_f$ 的成员 $f_Z$ 及任意正数 $\varepsilon$，均有 \[ \mathbb{P}\left\{ \sqrt{η_f}\, \sup_{t\in\mathbb{R}} \left| f_Z(t) - \Exf{f_Z(t)} \right| \ge \varepsilon\sqrt{U_{\mathcal{M}}(η_f)} \right\} \le 2\mathrm{e}^{-2\varepsilon^2} \] 成立的最小函数。我们证明对任意 $x> 1$，有 \[ \left| \sqrt{U_{\mathcal{M}}(x)} - \sqrt{\log_4 x} \right| \le 2 \min\!\left\{ 1,\, \frac{2 \ln(\mathrm{e} + \ln x)}{\sqrt{\ln x}} \right\}\,. \] 最优尺度 $\sqrt{U_{\mathcal{M}}(x)}$ 在有限样本量下一致地紧密关联于 $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$。