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

4.2 The canonical extremal measure

Theorem 10 Extremal measure saturating the tail cap
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Lean declarations

There exists an explicit probability measure \(\mu ^\star \) with mean zero such that

\[ \mathbb {P}_{\mu ^\star }(|x|{\gt}t) = s_G(t) \qquad (t \ge 0). \]

This is the canonical witness showing that the stop-loss envelope is sharp.

Proof ▶

Define a piecewise density whose right tail matches the sub-Gaussian cap and whose left side is adjusted so that the total mass is one and the mean vanishes. The right and left tail integrals match the desired formulas exactly, and the probability and mean-zero statements follow by direct integration.

Theorem 11 The extremizer attains the envelope
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Uses

Lean declarations

For every \(u \ge 0\),

\[ \mathrm{stopLoss}_{\mu ^\star }(u) = J_G(u). \]
Proof ▶

Given the tail identity from Theorem 10, the layer-cake formula turns the stop-loss of \(\mu ^\star \) into the integral of the tail cap itself. That integral is exactly the definition of the envelope on the tail branch, while the affine branch is forced by the centering condition. Thus the upper envelope from Chapter 3 is the true optimum.