A **discrete** distribution $p$ over $\mathbb{N}$ is said to be *log-concave* if it satisfies the following conditions:


1. The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p\_i > 0$ iff $a\leq i \leq b$.
2. for all $i\in\mathbb{N}$, $p\_i^2 \geq p\_{i-1}p\_{i+1}$.


(in the literature, condition (1) is sometimes forgotten). This is the discrete analogue of continuous log-concave densities, and includes many families of usual discrete distributions.


What I am looking for is a set of lecture notes, papers or more generally references that provides an exhaustive (or as comprehensive as possible) list of theorems and properties of discrete log-concave distributions. As for now, I am aware of [Devroye '87](http://link.springer.com/article/10.1007%2FBF02307716) and (part of) [An '97](http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1933), but not much more.



Reference on (discrete) log-concave probability distributions
There is a 67 page review from last year, [Log-concavity and strong log-concavity: a review](http://arxiv.org/abs/1404.5886), A. Saumard, J.A. Wellner (2014):



> 
> We review and formulate results concerning log-concavity and
>  strong-log-concavity in both discrete and continuous settings. We show
>  how preservation of log-concavity and strongly log-concavity on $\mathbb{R}$
>  under convolution follows from a fundamental monotonicity result of
>  Efron (1969). We provide a new proof of Efron’s theorem using the
>  recent asymmetric Brascamp-Lieb inequality due to Otto and Menz
>  (2013). Along the way we review connections between log-concavity and
>  other areas of mathematics and statistics, including concentration of
>  measure, log-Sobolev inequalities, convex geometry, MCMC algorithms,
>  Laplace approximations, and machine learning.
> 
> 
> 


This review contains many references, including some to older reviews and monographs. A few references are listed here, with hyperlinks:


1. [A universal generator for discrete log-concave distributions,](http://link.springer.com/content/pdf/10.1007/BF02243398.pdf)
W Hörmann (1994).
2. [A simple universal generator for continuous and discrete univariate T-concave distributions,](http://epub.wu.ac.at/1756/1/document.pdf) J. Leydold (2001).
3. [Preservation of log-concavity on summation](http://arxiv.org/abs/math/0502548), O. Johnson, C. Goldschmidt (2005).
4. [Log-concavity and the maximum entropy property of the Poisson
distribution,](http://arxiv.org/abs/math/0603647) O. Johnson (2006).
5. [On the entropy and log-concavity of compound Poisson measures,](http://arxiv.org/abs/0805.4112) O. Johnson, I. Kontoyiannis, M.
Madiman (2008).
6. [Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete
compound Poisson measures,](http://arxiv.org/abs/0912.0581) O. Johnson, I. Kontoyiannis, M.
Madiman (2009).
7. [Strong log-concavity is preserved by convolution,](https://www.stat.washington.edu/research/reports/2012/tr600.pdf) J.A. Wellner
(2010).
8. [Asymptotics of the discrete log‐concave maximum likelihood estimator
and related applications,](http://arxiv.org/abs/1107.3904) F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).


