Reference on (discrete) log-concave probability distributions
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).
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.