Estimates for the rapid decay of concentration functions of \(n\)-fold convolutions (Q1266772)

For a probability distribution \(F\) defined on the Bord \(\sigma\)-field of subsets of the real line \(R\), its concentration function is defined as \(Q(F,b) =\sup_xF\{[x,x+b]\}\) for any \(b\geq 0\). Let \(F^n\) be the \(n\)-fold convolution of \(F\). This paper deals with the classical question on the rate of decay of \(Q(F^n,b)\) as \(n\to\infty\). Some upper bounds are presented for \(Q(F^n, b)\) based on the estimates in the central limit theorem and some elementary properties of concentration functions. These results are refinements of some earlier results obtained by other authors and their proofs are also different from most of the previous papers, where the method of characteristic functions had been used extensively.