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2019 Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension
Adrien Saumard
Electron. Commun. Probab. 24: 1-8 (2019). DOI: 10.1214/19-ECP266


Bi-log-concavity of probability measures is a univariate extension of the notion of log-concavity that has been recently proposed in a statistical literature. Among other things, it has the nice property from a modelisation perspective to admit some multimodal distributions, while preserving some nice features of log-concave measures. We compute the isoperimetric constant for a bi-log-concave measure, extending a property available for log-concave measures. This implies that bi-log-concave measures have exponentially decreasing tails. Then we show that the convolution of a bi-log-concave measure with a log-concave one is bi-log-concave. Consequently, infinitely differentiable, positive densities are dense in the set of bi-log-concave densities for $ L_{p}$-norms, $p\in \left [1,+\infty \right ]$. We also derive a necessary and sufficient condition for the convolution of two bi-log-concave measures to be bi-log-concave. We conclude this note by discussing a way of defining a multi-dimensional extension of the notion of bi-log-concavity.


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Adrien Saumard. "Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension." Electron. Commun. Probab. 24 1 - 8, 2019.


Received: 10 April 2019; Accepted: 17 September 2019; Published: 2019
First available in Project Euclid: 1 October 2019

zbMATH: 1423.60033
MathSciNet: MR4017135
Digital Object Identifier: 10.1214/19-ECP266

Primary: 60E05, 60E15


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