Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
Abstract
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.
Related
- Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
- Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
- Feasible Frontiers for Sub-Gamma Envelopes: the Variance--Pole Trade-off for Infinitely Divisible Laws
- Linear Preservers of Infinitely Divisible Matrices
- Risk diversification for infinitely divisible distributions
BibTeX
@misc{chen2026directional,
title = {{Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws}},
author = {Yichuan Chen and Xin Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02590},
eprint = {2609.02590},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02590},
}
RIS
TY - UNPB
TI - Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
AU - Yichuan Chen
AU - Xin Wang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02590
UR - https://arxiv.org/abs/2609.02590
AB - 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.
ER -