Approximation of convolutions by accompanying laws under the existence of moments of low orders (Q1291974)

Assume that a one-dimensional probability distribution \(F(x)\) has a finite moment of order \(1+\beta\), with some \(0<\beta\leq 1\). Then it is shown that the rate of approximation of the \(n\)th convolution of \(F\) by accompanying laws is \(O(n^{-\gamma})\), where \(\gamma=\beta\) if \(0<\beta\leq 1/2\) and \(\gamma=1/2\) if \(1/2\leq \beta\leq 1\). Futhermore, if \(1/2<\beta<\infty\) and the second moment of \(F\) is infinite, then the rate of approximation is \(o(n^{-1/2})\).