The estimation of proximity of distribution of sequential sums of independent identically distributed random vectors

Let $F$ be a distribution on $\mathbb R^k$, $F_k^n$ – times convolution of $F$ with itself, $\mathscr L^k=\{B\in\mathbb R^k,B=[a_1,b_1]\times\dots\times[a_k,b_k]\}$.
It is proved that
$$ \sup_{B\in\mathscr L^k}|F^{n+1}\{B\}-F^n\{B\}|\le\frac{c(F)}{\sqrt n}, $$
where $c(F)$ depends on some characteristics of $F$.