Electronic Communications in Probability

Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension

Adrien Saumard

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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.

Article information

Electron. Commun. Probab., Volume 24 (2019), paper no. 61, 8 pp.

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

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Primary: 60E05: Distributions: general theory 60E15: Inequalities; stochastic orderings

bi-log-concavity isoperimetric constant log-concavity

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Saumard, Adrien. Bi-log-concavity: some properties and some remarks towards a multi-dimensional extension. Electron. Commun. Probab. 24 (2019), paper no. 61, 8 pp. doi:10.1214/19-ECP266. https://projecteuclid.org/euclid.ecp/1569895736

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  • [1] M. Bagnoli and T. Bergstrom, Log-concave probability and its applications, Econom. Theory 26 (2005), no. 2, 445–469.
  • [2] S. Bobkov, Extremal properties of half-spaces for log-concave distributions, Ann. Probab. 24 (1996), no. 1, 35–48.
  • [3] S. G. Bobkov and C. Houdré, Isoperimetric constants for product probability measures, Ann. Probab. 25 (1997), no. 1, 184–205.
  • [4] A. Colesanti, Log-concave functions, Convexity and concentration, IMA Vol. Math. Appl., vol. 161, Springer, New York, 2017, pp. 487–524.
  • [5] L. Dümbgen, P. Kolesnyk, and R. A. Wilke, Bi-log-concave distribution functions, J. Statist. Plann. Inference 184 (2017), 1–17.
  • [6] C. M. Fortuin, P. W. Kasteleyn, and J. Ginibre, Correlation inequalities on some partially ordered sets, Comm. Math. Phys. 22 (1971), no. 2, 89–103.
  • [7] O. Guédon, Concentration phenomena in high dimensional geometry, Proceedings of the Journées MAS 2012 (Clermond-Ferrand, France), 2012.
  • [8] N. Laha and J. A. Wellner, Bi-$s^{*}$-concave distributions, (2017), arXiv:1705.00252, preprint.
  • [9] M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, vol. 89, American Mathematical Society, Providence, RI, 2001.
  • [10] A. Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343.
  • [11] W. Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.
  • [12] R. J. Samworth, Recent progress in log-concave density estimation, Statist. Sci. 33 (2018), no. 4, 493–509.
  • [13] A. Saumard and J. A. Wellner, Log-concavity and strong log-concavity: A review, Statist. Surv. 8 (2014), 45–114.
  • [14] G. Walther, Inference and modeling with log-concave distributions, Statist. Sci. 24 (2009), no. 3, 319–327.