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

3.3 From stop-loss control to convex domination

Theorem 8 Convex domination from stop-loss comparison
✓
#

Uses

Lean declarations

If the Gaussian stop-loss transform dominates the envelope \(J_G\) pointwise, then every centered sub-Gaussian law is convex-dominated by the Gaussian comparator at that same scale.

Proof ▶

First prove the comparison for simple convex functions (finite sums of affine functions and hinges). Then approximate an arbitrary convex integrable function from below by a monotone sequence of such simple convex functions. Monotone convergence passes the inequality to the limit.