Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
Abstract
Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound for whose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2) multiplier give an exact variational formula for C_X. The resulting extended-valued function is positive homogeneous and is pointwise least among all homogeneous denominators compatible with the covariance quadratic form. We prove linear-map, convolution, and Levy-time rules, and show that the full family of directional remainders determines the law of X. Geometrically, C_X lies between the Minkowski functional of the moment-generating-function domain and one third of the positive support function of the Levy measure; the upper constant is sharp, and the zero set is a polar cone. The pole need not be subadditive. It is continuous on the sphere under global exponential moments and positive-definite covariance, whereas finite variance alone permits a jump from zero to infinity in nearby directions. For additive gamma-ray models, C_X equals the domain gauge and has a finite-polytope formula.
Related
- Feasible Frontiers for Sub-Gamma Envelopes: the Variance--Pole Trade-off for Infinitely Divisible Laws
- Directional Optimal Sub-Gamma Scales for Infinitely Divisible Laws
- Linear Preservers of Infinitely Divisible Matrices
- Risk diversification for infinitely divisible distributions
- Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
BibTeX
@misc{chen2026minimal,
title = {{Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors}},
author = {Yichuan Chen and Xin Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02595},
eprint = {2609.02595},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02595},
}
RIS
TY - UNPB
TI - Minimal Radial Sub-Gamma Envelopes for Infinitely Divisible Random Vectors
AU - Yichuan Chen
AU - Xin Wang
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02595
UR - https://arxiv.org/abs/2609.02595
AB - Let X be a centered infinitely divisible random vector with finite second moment and covariance matrix Sigma. We define the radial pole C_X(t) as the smallest scale in a right sub-gamma bound for whose quadratic proxy is fixed at the true variance t^T Sigma t. A canonical directional measure and an independent Beta(1,2) multiplier give an exact variational formula for C_X. The resulting extended-valued function is positive homogeneous and is pointwise least among all homogeneous denominators compatible with the covariance quadratic form. We prove linear-map, convolution, and Levy-time rules, and show that the full family of directional remainders determines the law of X. Geometrically, C_X lies between the Minkowski functional of the moment-generating-function domain and one third of the positive support function of the Levy measure; the upper constant is sharp, and the zero set is a polar cone. The pole need not be subadditive. It is continuous on the sphere under global exponential moments and positive-definite covariance, whereas finite variance alone permits a jump from zero to infinity in nearby directions. For additive gamma-ray models, C_X equals the domain gauge and has a finite-polytope formula.
ER -