Am I correct to interpret the following as (also) saying that if a sequence of log-concave densities converges in distribution, then it also converges in total variation? (Meaning that the weighted total variation from (c) is equal to total variation when $a=0$)
Proposition 2. Let $\left(f_{n}\right)$ be a sequence of log-concave densities on $\mathbb{R}^{d}$ with $f_{n} \stackrel{d}{\rightarrow} f$ for some density $f$. Then:
(a) $f$ is log-concave
(b) $f_{n} \rightarrow f, \mu$-almost everywhere
(c) Let $a_{0}>0$ and $b_{0} \in \mathbb{R}$ be such that $f(x) \leq e^{-a_{0}\|x\|+b_{0}}$. Then for every $a<a_{0}$, we have $\int_{\mathbb{R}^{d}} e^{a\|x\|}\left|f_{n}(x)-f(x)\right| d x \rightarrow 0$ and, if $f$ is continuous, $\sup _{x \in \mathbb{R}^{d}} e^{a\|x\|}\left|f_{n}(x)-f(x)\right| \rightarrow 0 .$
Reference: Cule, Madeleine, and Richard Samworth. "Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density." Electronic Journal of Statistics 4 (2010): 254-270.
