发表机构
Zhejiang Sci-Tech University（浙江理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文确定了中心化无穷可分律的亚伽马包络可行边界，给出精确变分公式，刻画前沿的凸性与下降轮廓，并分析其与Chernoff偏差的相等条件及代价。
AI 中文摘要
右亚伽马界由二次代理$v$和极点$c$描述，且该对不唯一：扩大任一者均保持其性质。将$v$固定为方差$V$可消除二次阶的歧义，但这是一种约定而非必然结果。对于具有有限非零方差的中心化无穷可分律，我们确定了可行集的完整边界，即映射$v\mapsto c_*(v)$。将Beta$(1,2)$乘子应用于归一化Kolmogorov正则测度，可将可行性转化为一维比较，并得出精确变分公式：前沿是凸的、非增的，且其可行集是凸的。矩生成函数的收敛横坐标对$c_*$施加了一个与$v$无关的下限，因此方差精确极点位于该下限上的律具有平坦前沿，而三阶累积量障碍仅存在于$v=V$处。每当该极点严格高于两个下限时，一旦$v>V$，前沿严格下降；若控制也是纯局部的且$\kappa_3^2/(9V^2)>\kappa_4/(12V)$，则下降具有平方根轮廓，而在其余边界情形下具有立方根轮廓，两种情况下在$V$处导数均为$-\infty$。随后我们刻画了优化前沿何时重现精确Chernoff偏差：当且仅当某对位于前沿上的点在某水平的Legendre最大化点处与累积量生成函数相切时，等式在该水平成立。在示例中列出的水平上，$v=V$的代价为百分之四到三十；对于中心化指数分布，每个水平上均存在差距，最高达百分之十三。
英文摘要
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 $κ_3^2/(9V^2)>κ_4/(12V)$, the drop has a square-root profile, and in the remaining boundary case a cube-root profile, with derivative $-\infty$ 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.
Comments27 pages
