Conditioning, weak limits and their distributions (Q1600586)

The fixed distribution \(p\) on \(\mathbb{R}\) with a finite first moment is considered. The classes of distributions related to \(p\) and defined by the subordination or by such notions as conditioning, weak limits and amost sure limits of arithmetic means are equal. The weak convergence in \(L_1\) of a sequence of random variables to a certain limit, together with the weak convergence of distributions implies a stochastic convergence. The weak convergence in \(L_1\) appears if other types of a convergence cannot be expected. The main ideas of formerly published results are explained for distributions \(p\) concentrated on \(\pm 1\) with values \(1/2\). Some unsolved problems are given.