We investigate the sub-Gaussian property for almost surely bounded random variables. If sub-Gaussianity per se is de facto ensured by the bounded support of said random variables, then exciting research avenues remain open. Among these questions is how to characterize the optimal sub-Gaussian proxy variance? Another question is how to characterize strict sub-Gaussianity, defined by a proxy variance equal to the (standard) variance? We address the questions in proposing conditions based on the study of functions variations. A particular focus is given to the relationship between strict sub-Gaussianity and symmetry of the distribution. In particular, we demonstrate that symmetry is neither sufficient nor necessary for strict sub-Gaussianity. In contrast, simple necessary conditions on the one hand, and simple sufficient conditions on the other hand, for strict sub-Gaussianity are provided. These results are illustrated via various applications to a number of bounded random variables, including Bernoulli, beta, binomial, Kumaraswamy, triangular, and uniform distributions.
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DOI : 10.1051/ps/2019018
Mots-clés : Sub-Gaussian, beta distribution, Kumaraswamy distribution, triangular distribution
@article{PS_2020__24_1_39_0,
author = {Arbel, Julyan and Marchal, Olivier and Nguyen, Hien D.},
title = {On strict {sub-Gaussianity,} optimal proxy variance and symmetry for bounded random variables},
journal = {ESAIM: Probability and Statistics},
pages = {39--55},
publisher = {EDP-Sciences},
volume = {24},
year = {2020},
doi = {10.1051/ps/2019018},
mrnumber = {4053001},
zbl = {1445.60016},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2019018/}
}
TY - JOUR AU - Arbel, Julyan AU - Marchal, Olivier AU - Nguyen, Hien D. TI - On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables JO - ESAIM: Probability and Statistics PY - 2020 SP - 39 EP - 55 VL - 24 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/ps/2019018/ DO - 10.1051/ps/2019018 LA - en ID - PS_2020__24_1_39_0 ER -
%0 Journal Article %A Arbel, Julyan %A Marchal, Olivier %A Nguyen, Hien D. %T On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables %J ESAIM: Probability and Statistics %D 2020 %P 39-55 %V 24 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/ps/2019018/ %R 10.1051/ps/2019018 %G en %F PS_2020__24_1_39_0
Arbel, Julyan; Marchal, Olivier; Nguyen, Hien D. On strict sub-Gaussianity, optimal proxy variance and symmetry for bounded random variables. ESAIM: Probability and Statistics, Tome 24 (2020), pp. 39-55. doi : 10.1051/ps/2019018. https://www.numdam.org/articles/10.1051/ps/2019018/
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