Let F be a symmetric k-dimensional probability distribution, whose characteristic function {Mathematical expression} satisfies for all t ∈R^k the inequality {Mathematical expression} ≥ -1 + α, where 0 < α < 2. Let n be an arbitrary natural number, let Fⁿ be the n-fold convolution of the distribution F with itself, and let e(nF) be the accompanying infinitely divisible distribution with characteristic function exp(n( {Mathematical expression} -1)). It is proved that the uniform distance ρ(·,·) between corresponding distribution functions admits estimate ρ(Fⁿ,e(nF))≤c₁(k)(n^-1+exp(-nα+c_ℓkℓn³n)), where c₁(k) depends only on the dimension k, while c₂is an absolute constant.