AI 中文总结
该研究针对无穷可分律，通过归一化Kolmogorov典范测度并结合Beta(1,2)变量得到带符号余项，推导了定向最优次伽马尺度的公式，明确了其性质并应用于双侧伽马律与中心化Skellam律。
AI 中文摘要
将次伽马界中的二次代理固定为真实方差时，会留下一个待优化的尺度，而双侧无穷可分律通常在两个方向上需要不同的尺度。对于具有有限非零方差的中心化律，我们对其Kolmogorov典范测度进行归一化，并将所得变量乘以独立的Beta(1,2)变量。由此得到的带符号余项给出了左右尺度的精确变分公式。我们证明，当Levy测度在某一方向上无跳跃时，该方向的定向尺度恰好为零；建立了反射、缩放、卷积、Levy时间及反向跳跃扰动规则，并从余项律中恢复出Levy三元组。这些公式给出了双侧伽马律的两个伽马尺度，对于中心化Skellam律，给出了局部与内部控制之间的精确过渡点p₊=(2+√3)/4和p₋=(2-√3)/4；当正跳跃稀少时，右侧尺度渐近于1/log(1/p)。因此，该对记录了跳跃方向与控制方差的机制——精确的次伽马极点。
英文摘要
Fixing the quadratic proxy in a sub-gamma bound at the true variance leaves a scale to optimize, and a two-sided infinitely divisible law generally requires different scales in the two directions. For a centered law with finite nonzero variance, we normalize its Kolmogorov canonical measure and multiply the resulting variable by an independent Beta(1,2) variable. The signed remainder obtained in this way gives exact variational formulas for the right and left scales. We prove that a directional scale vanishes exactly when the Levy measure has no jumps in that direction, establish reflection, scaling, convolution, Levy-time, and opposite-jump perturbation rules, and recover the Levy triplet from the remainder law. The formulas give the two Gamma scales for bilateral Gamma laws and, for centered Skellam laws, the exact transition points p_+=(2+sqrt(3))/4 and p_-=(2-sqrt(3))/4 between local and interior control; when positive jumps are rare, the right scale is asymptotic to 1/log(1/p). The pair therefore records jump direction and the mechanism that controls the variance-exact sub-gamma pole.
Comments25 pages
