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$\mathbb{R}$ with values in $[0,1]$, and let
$U_{\mathcal{M}}\colon \mathbb{R}_+\to \mathbb{R}_+$ be the minimal
function for which
$$
\mathbb{P}\left\{ \sqrt{\eta_f}\, \sup_{t\in\mathbb{R}}
\left| f_Z(t) - \Exf{f_Z(t)} \right|
\ge \varepsilon\sqrt{U_{\mathcal{M}}(\eta_f)} \right\}
\le 2\e^{-2\varepsilon^2}
$$
holds for every member $f_Z$ of $\mathcal{M}$ with finite effective sample size
$\eta_f$ 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.
From: Rabee Tourky [view email]
[v1]
Sun, 3 Dec 2023 03:20:09 UTC (30 KB)
[v2]
Sun, 10 Dec 2023 09:36:24 UTC (31 KB)
[v3]
Fri, 22 Dec 2023 04:36:10 UTC (32 KB)
[v4]
Thu, 4 Jun 2026 01:10:33 UTC (68 KB)
[v5]
Mon, 15 Jun 2026 00:30:20 UTC (77 KB)
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