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:

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

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

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

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

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

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

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

  8. 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.