Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws

Abstract

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.

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BibTeX RIS

BibTeX

@misc{chen2026optimal,
  title = {{Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws}},
  author = {Yichuan Chen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.00032},
  eprint = {2610.00032},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.00032},
}

RIS

TY  - UNPB
TI  - Optimal Sub-Gamma Scales for Spectrally Positive Infinitely Divisible Laws
AU  - Yichuan Chen
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.00032
UR  - https://arxiv.org/abs/2610.00032
AB  - 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.
ER  -

Abstract

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