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

@misc{chen2026feasible,
  title = {{Feasible Frontiers for Sub-Gamma Envelopes: the Variance--Pole Trade-off for Infinitely Divisible Laws}},
  author = {Yichuan Chen and Xin Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23069},
  eprint = {2609.23069},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23069},
}

TY  - UNPB
TI  - Feasible Frontiers for Sub-Gamma Envelopes: the Variance--Pole Trade-off for Infinitely Divisible Laws
AU  - Yichuan Chen
AU  - Xin Wang
PY  - 2026
DA  - 2026/09/19
PB  - arXiv
DO  - 10.48550/arxiv.2609.23069
UR  - https://arxiv.org/abs/2609.23069
AB  - 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↦ 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 κ_3^2/(9V^2)>κ_4/(12V), the drop has a square-root profile, and in the remaining boundary case a cube-root profile, with derivative −∞ 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.
ER  -