Reference on (discrete) log-concave probability distributions A discrete distribution $p$ over $\mathbb{N}$ is said to be log-concave if it satisfies the following conditions:

*

*The support of $p$ is a contiguous interval, i.e. $\exists a \leq b$ s.t. $p_i > 0$ iff $a\leq i \leq b$.

*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 and (part of) An '97, but not much more.
 A: There is a very nice 36-page review on log-concavity and unimodality in the discrete setting by Richard Stanley, published in 1989. It is titled "Log-concave and unimodal sequences in algebra, combinatorics, and geometry" and available online here:
http://onlinelibrary.wiley.com/doi/10.1111/j.1749-6632.1989.tb16434.x/abstract
As one might expect from the title, this survey does not focus as much on log-concave probability distributions on the positive integers; however, there are many useful things that one can learn here in any case (imposing the requirement of the sum of the sequence being 1 if necessary).
There is also the notion of ultra-log-concavity (discussed briefly in the Saumard-Wellner survey mentioned in the previous answer); this has beautiful connections not just to probability (where it can be interpreted as relative log-concavity with respect to binomial or Poisson distributions), but also to combinatorics. For recent papers that utilize this notion, see for example:
Kahn and Neiman: "A strong log-concavity property for measures on Boolean algebras", JCT(A), 2011.
and
Nayar and Oleszkiewicz: "Khinchine type inequalities with optimal constants via ultra log-concavity", Positivity, 2012.
A: There is a 67 page review from last year, Log-concavity and strong log-concavity: a review, 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:


*

*A universal generator for discrete log-concave distributions,
W Hörmann (1994).

*A simple universal generator for continuous and discrete univariate T-concave distributions, J. Leydold (2001).

*Preservation of log-concavity on summation, O. Johnson, C. Goldschmidt (2005).

*Log-concavity and the maximum entropy property of the Poisson
distribution, O. Johnson (2006).

*On the entropy and log-concavity of compound Poisson measures, O. Johnson, I. Kontoyiannis, M.
Madiman (2008).

*Log-concavity, ultra-log-concavity, and a maximum entropy property of discrete
compound Poisson measures, O. Johnson, I. Kontoyiannis, M.
Madiman (2009).

*Strong log-concavity is preserved by convolution, J.A. Wellner
(2010).

*Asymptotics of the discrete log‐concave maximum likelihood estimator
and related applications, F. Balabdaoui, H. Jankowski, K. Rufibach, M. Pavlides, (2011).
A: J Pitman's 1997 article, https://www.stat.berkeley.edu/~pitman/453.pdf, deals with a condition slightly stronger than log concave, PF (Polya frequency), and has a lot of fascinating results and estimates. 
