On large deviations for the sum of nonidentically distributed random variables

Let $X_1,X_2,\dots$ be independent random variables such that $\mathbf EX_i=0$, $\mathbf EX_i^2<\infty$ ($i\ge 1$) and for every $k=1,2,\dots$
$$ B_{k,n}^2=\mathbf EX_{k+1}^2+\dots+\mathbf EX_{k+n}^2\to\infty\qquad(n\to\infty). $$
We obtain necessary and sufficient conditions for the relations
$$ \mathbf P\{X_{k+1}+\dots+X_{k+n}\ge xB_{k,n}\}=[1-\Phi(x)][1+\varepsilon(B_{n,k})] $$
to hold uniformly for $x\in[0,\Lambda(B_{k,n}^2)]$ and $k=1,2,\dots$, where $\Phi(x)$ is a standard normal distribution function, $\varepsilon(t)\to 0\,(t\to\infty)$, $\Lambda(t)$ is a nonnegative monotone function with properties (3) or $\Lambda(t)=c\sqrt{\ln t},\,c>0$.