Convergence of the local limit theorem in $L^1$

Question

Let $(X_i)_{i \in \mathbb{N}}$ be a sequence of iid random variables with common density $f$ with expectation $0$ and variance $\sigma$. Set $f_n$ to be the density of the sum \begin{align*} S_n = \frac{X_1+...+X_n}{\sqrt{n}} \end{align*} Let $\phi$ be the density of a normally distributed random variable with expectation $0$ and variance $\sigma$. The local limit theorem now deals with the convergence of $f_n$ to $\phi$.

I am trying to find conditions on $f$ such that $f_n$ converges in $L^1$ to $\phi$. Do you know such conditions, and also some literature where I can read up on that?

Answer 1

If you have an $f$, such that $f_n \to \phi$ pointwise then $f_n \to \phi$ in $L^1$. It is a consequence of Scheffé's theorem.