Feasible Frontiers for Sub-Gamma Envelopes: the Variance--Pole Trade-off for Infinitely Divisible Laws

Abstract

A right sub-gamma bound is described by a quadratic proxy $v$ and a pole $c$, and the pair is not unique: enlarging either preserves it. Fixing $v$ at the variance $V$ removes the ambiguity at quadratic order but is a convention, not a consequence. For centered infinitely divisible laws with finite nonzero variance we determine the entire boundary of the feasible set, the map $v\mapsto c_*(v)$. A Beta$(1,2)$ multiplier applied to the normalised Kolmogorov canonical measure turns feasibility into a one-dimensional comparison and yields an exact variational formula: the frontier is convex, nonincreasing, and its feasible set is convex. The abscissa of convergence of the moment generating function imposes a $v$-insensitive floor on $c_*$, so a law whose variance-exact pole sits on it has a flat frontier, whereas the third-cumulant obstruction exists only at $v=V$. Whenever that pole lies strictly above both floors the frontier drops strictly as soon as $v>V$; if the control is also purely local with $\kappa_3^2/(9V^2)>\kappa_4/(12V)$, the drop has a square-root profile, and in the remaining boundary case a cube-root profile, with derivative $-\infty$ at $V$ either way. We then characterise when the optimised frontier reproduces the exact Chernoff deviation: equality holds at a level exactly when some pair on the frontier is tangent to the cumulant generating function at a Legendre maximiser for that level. At the levels tabulated in a worked example $v=V$ costs four to thirty percent; for a centered exponential a gap persists at every level, reaching thirteen percent.