## Electronic Communications in Probability

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

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

#### Abstract

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

**Source**

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

**Dates**

Received: 10 April 2019

Accepted: 17 September 2019

First available in Project Euclid: 1 October 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1569895736

**Digital Object Identifier**

doi:10.1214/19-ECP266

**Subjects**

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

**Keywords**

bi-log-concavity isoperimetric constant log-concavity

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

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