Abstract
We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, thus proving upper bounds recently conjectured by Elder (2016). We provide different proof techniques for the symmetrical (around its mean) case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function. As a consequence, we derive the optimal proxy variance for the Dirichlet distribution, which is apparently a novel result. We also provide a new proof of the optimal proxy variance for the Bernoulli distribution, and discuss in this context the proxy variance relation to log-Sobolev inequalities and transport inequalities.
Highlights
The sub-Gaussian property (Buldygin and Kozachenko, 1980, 2000; Pisier, 2016) and related concentration inequalities (Boucheron et al, 2013; Raginsky and Sason, 2013) have attracted a lot of attention in the last couple of decades due to their applications in various areas such as pure mathematics, physics, information theory and computer sciences
We obtain the optimal proxy variance for the sub-Gaussianity of Beta distribution, proving upper bounds recently conjectured by Elder (2016)
We provide different proof techniques for the symmetrical case and the non-symmetrical case. The technique in the latter case relies on studying the ordinary differential equation satisfied by the Beta moment-generating function known as the confluent hypergeometric function
Summary
The sub-Gaussian property (Buldygin and Kozachenko, 1980, 2000; Pisier, 2016) and related concentration inequalities (Boucheron et al, 2013; Raginsky and Sason, 2013) have attracted a lot of attention in the last couple of decades due to their applications in various areas such as pure mathematics, physics, information theory and computer sciences. Applying the result of Bobkov and Götze (1999) to the T. transport inequality (1.5) yields a distribution-sensitive proxy variance of for the Bernoulli with mean μ. Transport inequality (1.5) yields a distribution-sensitive proxy variance of for the Bernoulli with mean μ It is optimal, see for instance Theorem 3.4.6 of Raginsky and Sason (2013). See for instance Theorem 3.4.6 of Raginsky and Sason (2013) This viewpoint highlights the key role played by the balance coefficient in the non-uniformity of the optimal proxy variance for discrete distributions such as the Bernoulli. The R code for the plots presented in this note and for a function deriving the optimal proxy variance in terms of α and β is available at http://www.julyanarbel.com/software
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