Approximation of convolutions by accompanying laws under the existence of moments of low orders

Abstract

It is shown that if a one-dimensional distribution F has finite moment of order 1+β for some β, 1/2≤β≤1, then the rate of approximation of the n-fold convolution Fn by accompanying laws is O(n−1/2). Futhermore, if Eξ2 = ∞ and 1/2<β<1, then the rate of approximation is o(n−1/2). The question about the true rate of approximation of Fn by infinitely divisible and accompanying laws is discussed. Bibliography: 27 titles.