On asymptotic behaviour of the remainder term in the central limit theorem

Let $X_1,X_2,\dots$ be a sequence of independent identically distributed random variables with zero means and unit variances. Let $k_n$ be a sequence of natural numbers, $k_n\to\infty$, $k_{n+1}/k_n\to 1$ ($n\to\infty$),
$$ F_n(x)=\mathbf P\biggl\{\frac{1}{\sqrt{k_n}}\sum_{k=1}^{k_n}X_k<x\biggr\},\qquad \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt. $$
We study conditions under which
$$ F_n(x)=\Phi(x)+\frac{\Psi(x)+o(1)}{\mu_n}\qquad (n\to\infty) $$
uniformly in $x$, $-\infty<x<\infty$, where $\mu_n$ is a positive sequence such that $\mu_n\to\infty$, $\mu_n=o(k_n)$ ($n\to\infty$).