Approximation of convolutions of multi-dimensional symmetric distributions by accompanying laws

Abstract

Let F be a symmetric k-dimensional probability distribution, whose characteristic function$\hat F(t)$ satisfies for allt ∈R ^k the inequality$\hat F(t)$ ≥ −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($\hat F(t)$ −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_ℓkℓn ³ n)), where c₁ (k) depends only on the dimension k, while c₂ is an absolute constant.