Convex Order Comparisons for Sub-Gamma Random Variables
Abstract
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∨ 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∨ α, but is attained only when α≤ 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.
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BibTeX
@misc{khribch2026convex,
title = {{Convex Order Comparisons for Sub-Gamma Random Variables}},
author = {El Mahdi Khribch and Badr-Eddine Chérief-Abdellatif},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02398},
eprint = {2609.02398},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02398},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Convex Order Comparisons for Sub-Gamma Random Variables
AU - El Mahdi Khribch
AU - Badr-Eddine Chérief-Abdellatif
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02398
UR - https://arxiv.org/abs/2609.02398
AB - 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∨ 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∨ α, but is attained only when α≤ 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.
ER -