Convergence of convolutions of concentration functions to degenerate, normal, and Poisson laws

The paper compares the behavior of convolutions of distribution functions and that of convolutions of corresponding concentration functions. It is shown that the weak convergence of a sequence of convolutions of distribution functions is equivalent to the weak convergence of a sequence of convolutions of corresponding concentration functions to normal, Poisson, and degenerate laws. The most of statements are supposed to be uniform, inside the convolution, closeness of the concentration components to degenerate laws.