次伽马随机变量的凸序比较
Convex Order Comparisons for Sub-Gamma Random Variables
- ESSEC Business School（ESSEC商学院）
- CNRS, LPSM, Sorbonne Université, Université Paris Cité（法国国家科学研究中心，LPSM，索邦大学，巴黎西岱大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究次伽马随机变量的凸序比较问题，推导最优倍数的变分公式并证明其精确性，同时分析次指数类的不同行为，依赖Kearns-Saul不等式的有限范围变体得到相关结果。
AI中文摘要：
近期研究表明，次高斯随机变量在凸序中被高斯分布的某个精确倍数所控制。本文研究基于伯恩斯坦不等式的次伽马类的类似问题，以拉普拉斯分布作为控制分布。此处矩生成函数由有限频率范围内的伯恩斯坦型界控制，而非纯二次界。我们推导了最优倍数的变分公式，证明其严格大于自然尺度σ∨α，并证明该值是精确的，可由非对称两点分布达到。随后转向次指数类，其与次高斯类类似具有二次界，但仅在有限范围内成立。有趣的是，次指数类表现出不同的行为：最优倍数恰好为σ∨α，但仅当α≤σ时达到；当α>σ时，该常数仍精确，但对任何非仿射凸函数无法达到等式。两个结果均依赖于Kearns-Saul不等式的精确有限范围变体，该变体本身具有独立研究价值。
英文摘要：
Recent work has shown that sub-Gaussian random variables are dominated in convex order by a sharp multiple of a Gaussian. We study the analogous question for the sub-Gamma class underlying Bernstein's inequality, with the Laplace law as the majorant. Here the moment generating function is controlled by a Bernstein-type bound over a finite range of frequencies, rather than by a purely quadratic bound. We derive a variational formula for the optimal multiple, show that it is strictly larger than the natural scale $σ\vee α$, and prove that this value is sharp, being attained by an asymmetric two-point distribution. We then turn to the sub-exponential class, which has a quadratic bound as in the sub-Gaussian case, but only over a bounded range as in the sub-Gamma case. Interestingly, the sub-exponential class displays a different behavior: the optimal multiple is exactly $σ\vee α$, but is attained only when $α\leq σ$; when $α> σ$, the constant remains sharp, but equality cannot hold for any non-affine convex function. Both results rely on a sharp finite-range variant of the Kearns-Saul inequality, which is of independent interest.
