log-concave distribution

Definition

A measure $\mu$ is log-concave if for all measurable sets $A,B$ and all $\lambda \in [0,1]$, we have

$$\mu (\lambda A+(1-\lambda )B)\ge \mu (A{)}^{\lambda }\mu (B{)}^{1-\lambda },$$

if the set $\lambda A+(1-\lambda )B$ is measurable.

Equivalently, if if a density $f$ can be written as $f(x)=\exp (\phi (x))$ for some concave $\phi$, then the distribution is log-concave.

If $X$ is a log-concave distributed random vector, then we can state a bound on its ${\psi }_1$ Orlicz norm, thus demonstrating that it is sub-exponential. In particular, Lemma 2.3 [here] (see also Equation (21) here) demonstrate that

$${\Vert ⟨u,X-\mathbb{E}X⟩\Vert }_{{\psi }_1}\le c\sqrt{\mathbb{E}⟨u,X-\mu {⟩}^2}=c\sqrt{⟨u,\Sigma u⟩},$$

for some $c\ge 0$ and all unit vectors $u$ where $\Sigma =\mathbb{V}(X)$. This demonstrates that all log-concave distributions are sub-exponential.

Many common distributions are log-concave:

• The normal distribution
• Exponential distribution
• Dirichlet distribution for parameters all at least 1
• Laplace distribution
• and others

Time-uniform concentration inequalities for log-concave vectors are given in multivariate light-tailed concentration.

Refs

• https://sites.stat.washington.edu/jaw/RESEARCH/TALKS/Toulouse1-Mar-p1-small.pdf