Moderate deviations for densities in $\mathbb R^k$

Let $\{X^i=(X^i_1,\dots,X^i_k),\, i=1,2,\dots\}$ be a sum of independent identically distributed random vectors in $\mathbb R^k$, let $\varphi_{0,E}$ be a density of standard normal distribution in $\mathbb R^k$ and let $p_n(x)$ be a density of $z_n=\frac{1}{\sqrt n}S_n$. Conditions upon distribution of $X^1$ are given under which
$$ p_n(x)=\varphi_{0,E}(x)(1+o(1)),\quad n\to\infty $$
uniformly in $x$, $x\in\mathcal A_n$, and $\mathcal A_n=\{x\in\mathbb R^k:|x|\leqslant c\sqrt{\log n}\}, c>0$.