谱正无穷可分律的最优子伽马尺度
Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
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中文总结 AI 辅助
该研究为谱正无穷可分律确定右端子伽马界的最优尺度，通过变分公式和矩生成函数比较，得到精确结果，并指出最优尺度可能位于不同位置。
中文摘要 AI 辅助
设X是一个中心化的谱正无穷可分随机变量，具有有限非零方差V。我们确定了右端子伽马界中的最小尺度c，其二次代理固定为V。归一化Kolmogorov典范测度的概率变换产生一个非负变量R_X，使得log E e^{tX} = V t^2 M_{R_X}(t)/2。因此，尺度问题归结为比较R_X的矩生成函数与指数律的矩生成函数，从而得到一个精确的一维变分公式。若X非高斯且具有正指数矩，则最优尺度至少为kappa_3(X)/(3V)，当且仅当R_X属于Klar和Mueller的M类时取等号。标准子伽马包络中的等式仅在高斯律或非高斯情形下具有指数跳跃的中心化复合泊松律中成立。对于一般的复合泊松律，R_X是跳跃大小的二阶平衡分布。对伽马和有界两点跳跃的精确计算表明，最优值可能出现在原点、正矩生成函数边界或内点处。
英文摘要
Let X be a centered spectrally positive infinitely divisible random variable with finite nonzero variance V. We determine the smallest scale c in a right-sided sub-gamma bound whose quadratic proxy is fixed at V. A probability transform of the normalized Kolmogorov canonical measure produces a nonnegative variable R_X for which log E e^{tX} = V t^2 M_{R_X}(t)/2. The scale problem therefore reduces to comparing the MGF of R_X with that of an exponential law, leading to an exact one-dimensional variational formula. If X is non-Gaussian and has a positive exponential moment, the optimal scale is at least kappa_3(X)/(3V), with equality precisely when R_X belongs to the M-class of Klar and Mueller. Equality in the standard sub-gamma envelope occurs only for the Gaussian law or, in the non-Gaussian case, for centered compound Poisson laws with exponential jumps. For a general compound Poisson law, R_X is the second-order equilibrium distribution of the jump size. Exact calculations for gamma and bounded two-point jumps show that the optimum may occur at the origin, at the positive MGF boundary, or at an interior point.
发表机构
- School of Art and Design, Zhejiang Sci-Tech University（浙江理工大学艺术与设计学院）
机构由 AI 辅助整理，请以论文原文为准。
