Sharp universal support criterion for sub-Gaussian laws
For every Borel probability law on the real line that is symmetric about zero and satisfies 0 < a <= |X| <= b almost surely, the support ratio b/a <= sqrt(3) + sqrt(2) guarantees E exp(t X) <= exp(t^2 E X^2 / 2) for every real t. The inequality is strict for nonzero t, and the actual variance is the optimal quadratic proxy. An explicit symmetric law on {-R,-1,1,R} violates the bound near zero for each larger ratio R. In the normalization 3-sqrt(6) <= X^2 <= 3+sqrt(6), the development also establishes the sharp eighth-order MGF gap with its optimal interiority coefficient, identifies the…
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