Reference request: local limit theorem for log-concave densities

Question

The following is easy to prove and can't possibly be new. But I can't find it printed anywhere despite some effort. Can anyone tell me where it is published?

Let $X_1,X_2,\ldots$ be a sequence of real 1-D random variables with continuous densities $f_1,f_2,\ldots$. Let $\phi(x)$ be the standard normal density. We are given that $$\limsup_{u\in\mathbb R}\,\Bigl|\,\int_{-\infty}^u f_n(x)\,dx - \int_{-\infty}^u \phi(x)\,dx\,\Bigr| \to 0 \text{ as } n\to\infty.$$ It does not necessarily follow that $$\limsup_{u\in\mathbb R}\, \bigl|\, f_n(u)-\phi(u)\bigr|\to 0 \text{ as } n\to\infty,~~~~~~~~(1)$$ but the theorem is that (1) holds if each $f_n$ is log-concave.

An analogous result was published by Bender for integer-valued random variables, but I need the continuous case.

Answer 1

If I understand correctly, then Section 2 (and especially Proposition 2) of the paper "Theoretical properties of the log-concave maximum likelihood estimator of a multidimensional density" by Madeleine Cule and Richard Samworth (2010), may be of use to you.

The assumptions of proposition 2 are convergence in distribution of the corresponding (to the densities) measures, and log-concave densities. Then they prove that the sequence of densities converges to the corresponding density "$\mu$-almost everywhere". I have not checked the proof.