Estimates for the stability of decompositions of three-point distributions
Let \(\rho(F,G)= \sup_x| F(x)- G(x)|\) denote the uniform (Kolmogorov) distance of two distribution functions \(F\) and \(G\). The main result of this paper seems to be following: For any sequence of convolutions \(F_n= F_{n1}* F_{n2}\) and any discontinuous distribution function \(G\) satisfying \(\rho (F_n,G)\to 0\) as \(n\to \infty\) the relation \[ \inf_{G_{n1} *G_{n2}=G} \max_{j=1, 2} \rho(F_{nj},G_{nj}) \to 0\quad\text{as }n\to \infty \] holds.
scientific article; zbMATH DE number 1296184