2.1 The comparison problem

We formalize the following question: what is the smallest positive constant \(c\) such that every centered real random variable \(X\) satisfying

\[ \mathbb {P}(|X| {\gt} t) \le 2 e^{-t^2/2} \qquad (t \ge 0) \]

is dominated in convex order by the centered Gaussian law of standard deviation \(c\)?

Definition 1 Admissible scale and sharp constant

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A positive number \(c\) is admissible if every centered real random variable \(X\) with the two-sided sub-Gaussian tail bound above satisfies

\[ \mathbb {E}[f(X)] \le \mathbb {E}[f(cG)] \]

for every convex function \(f\) whose expectations on both sides are finite. The sharp constant \(c_\star \) is the infimum of all admissible scales.

Definition 2 Explicit constants

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The explicit constant \(c_0\) is built from the sub-Gaussian tail cap

\[ s_G(t) := \min \{ 1, 2e^{-t^2/2}\} , \qquad t_0 := \sqrt{2 \log 2}, \]

the total tail mass

\[ A := \int _0^\infty s_G(t)\, dt, \qquad B := A/2, \]

the unique threshold \(a {\gt} t_0\) solving

\[ H(a) = a s_G(a) + \int _a^\infty s_G(t)\, dt = B, \]

the height \(p_0 := s_G(a)\), the Gaussian tail quantile \(z\) satisfying \(\overline{\Phi }(z) = p_0\), and finally

\[ c_0 := \frac{B}{\varphi (z)}. \]