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Exact bounds on the distribution function of isotropic log-concave distributions

  • Iosif Pinelis
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Abstract

For each real b, exact upper and lower bounds on the probability P(X≥ b) over all random variables X with log-concave p.d.f.'s such that E X=0 and E X2=1 are obtained, as well as the best constant factor C in the inequality P(X≥ b)≤ C e−b for all real b≥0. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.

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Cite

BibTeXRIS
BibTeX
@misc{pinelis2026exact,
  title = {{Exact bounds on the distribution function of isotropic log-concave distributions}},
  author = {Iosif Pinelis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10289},
  eprint = {2610.10289},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10289},
}
RIS
TY  - UNPB
TI  - Exact bounds on the distribution function of isotropic log-concave distributions
AU  - Iosif Pinelis
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2610.10289
UR  - https://arxiv.org/abs/2610.10289
AB  - For each real b, exact upper and lower bounds on the probability P(X≥ b) over all random variables X with log-concave p.d.f.'s such that E X=0 and E X^2=1 are obtained, as well as the best constant factor C in the inequality P(X≥ b)≤ C e^−b for all real b≥0. Exact upper and lower bounds on the corresponding exponential moments (Proposition 9.4) and explicit exponentially decreasing upper bounds on the mentioned p.d.f.'s (Propositions 9.2 and 9.3) are given as well. Some general results concerning log-concave p.d.f.'s are also obtained.
ER  -