Let $S_n=X^{(1)}+\dots+X^{(n)}$ be a sum of independent identically distributed random vectors in $R^k$ and let $\Phi$ be the standard normal distribution in $R^k$ . Conditions upon distribution of $X^{(1)}$ are given under which $$ \mathbf P\{S_n/\sqrt n\in A_n\}=\Phi(A_n)(1+o(1)),\quad n\to\infty, $$uniformly in sequences of Borel sets$\{A_n\}$such that$\Phi(A_n)\ge\Phi(x\colon|x|>\Lambda(n))$where$\Lambda(z)\uparrow\infty$is a function satisfying condition (8). In Theorems 1 and 2, we consider the case$\Lambda(z)=bz^\alpha$,$b>0$,$0<\alpha<1/2$.