The sharp one-dimensional convex sub-Gaussian comparison constant — Blueprint

2.2 Comparison and sharpness statements

The sharp theorem splits into two halves:

Theorem 3 Domination and sharpness
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  1. Domination. Every centered sub-Gaussian random variable is convex-dominated by the Gaussian law at scale \(c_0\).

  2. Sharpness. For every \(c {\lt} c_0\), domination fails: the canonical witness measure \(\mu ^\star \) and a hinge test function \(x \mapsto (x-u)_+\) exhibit a strict inequality in the opposite direction.

Proof

For the domination half, the proof pushes a random variable \(X\) forward to its law, verifies that the pushforward inherits the mean-zero and tail hypotheses, and then applies the measure-level domination result from the internal analytic argument (Theorem 8). For sharpness, the extremal witness \(\mu ^\star \) is constructed explicitly and its stop-loss transform is shown to exceed the Gaussian stop-loss at the threshold \(u = cz\).