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Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension. (English) Zbl 1423.60033
Summary: 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, $$p\in [1,+\infty ]$$. 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.
##### MSC:
 6e+06 Probability distributions: general theory 6e+16 Inequalities; stochastic orderings
##### Keywords:
bi-log-concavity; isoperimetric constant; log-concavity
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