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 $$ \mathbb{P}\left\{ \sqrt{η_f}\, \sup_{t\in\mathbb{R}} \left| f_Z(t) - \Exf{f_Z(t)} \right| \ge \varepsilon\sqrt{U_{\mathcal{M}}(η_f)} \right\} \le 2\e^{-2\varepsilon^2} $$ holds for every member $f_Z$ of $\mathcal{M}$ with finite effective sample size $η_f$ and every positive $\varepsilon$. We prove that for every $x> 1$, $$ \left| \sqrt{U_{\mathcal{M}}(x)} - \sqrt{\log_4 x} \right| \le 2 \min\!\left\{ 1,\, \frac{2 \ln(\e + \ln x)}{\sqrt{\ln x}} \right\}\,. $$ The optimal adjustment $\sqrt{U_{\mathcal{M}}(x)}$ matches $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$ for all $x>1$, with residuals bounded as above.

翻译：令 $\mathcal{M}$ 表示 $\mathbb{R}$ 上取值于 $[0,1]$ 的随机单调函数类，并设 $U_{\mathcal{M}}\colon \mathbb{R}_+\to \mathbb{R}_+$ 是满足如下性质的最小函数：对 $\mathcal{M}$ 中任意具有有限有效样本量 $η_f$ 的成员 $f_Z$ 及任意正数 $\varepsilon$，有 $$ \mathbb{P}\left\{ \sqrt{η_f}\, \sup_{t\in\mathbb{R}} \left| f_Z(t) - \Exf{f_Z(t)} \right| \ge \varepsilon\sqrt{U_{\mathcal{M}}(η_f)} \right\} \le 2\e^{-2\varepsilon^2}. $$ 我们证明：对所有 $x>1$， $$ \left| \sqrt{U_{\mathcal{M}}(x)} - \sqrt{\log_4 x} \right| \le 2 \min\!\left\{ 1,\, \frac{2 \ln(\e + \ln x)}{\sqrt{\ln x}} \right\}\,. $$ 最优调整项 $\sqrt{U_{\mathcal{M}}(x)}$ 对所有 $x>1$ 与 $\frac{1}{\sqrt{2\ln 2}}\sqrt{\ln x}$ 相匹配，且残差如上界所述。