A fast Berry-Esseen theorem under minimal density assumptions

Let $X_1,\ldots,X_N$ be i.i.d.\ random variables distributed like $X$. Suppose that the first $k \geq 3$ moments $\{ \mathbb{E}[X^j] : j = 1,\ldots,k\}$ of $X$ agree with that of the standard Gaussian distribution, that $\mathbb{E}[|X|^{k+1}] < \infty$, and that there is a subinterval of $\mathbb{R}$ of width $w$ over which the law of $X$ has a density of at least $h$. Then we show that \begin{align} \label{eq:bnew} \sup_{s \in \mathbb{R}} \left| \mathbb{P} \left( \frac{X_1 + \ldots + X_N}{ \sqrt{N} } \leq s \right) - \int_{-\infty}^s \frac{ e^{ - u^2/2} \mathrm{d} u }{ \sqrt{2 \pi }} \right| \leq 3 \left\{ \frac{\mathbb{E}[|X|^{k+1}]}{ N^{ \frac{k-1}{2}} } + e^{ - c hw^3 N/\mathbb{E}[|X|^{k+1}] } \right\}, \end{align} where $c > 0$ is universal. By setting $k=3$, we see that in particular all symmetric random variables with densities and finite fourth moment satisfy a Berry-Esseen inequality with a bound of the order $1/N$. Thereafter, we study the Berry-Esseen theorem as it pertains to perturbations of the Bernoulli law with a small density component, showing by means of a reverse inequality that the power $hw^3$ in the exponential term is asymptotically sharp.