The central limit theorem for dependent random variables

The central limit theorem has been extended to the case of dependent random variables by several authors (Bruns, Markoff, S. Bernstein, P. Levy, Loeve). The conditions under which these theorems are stated either are very restrictive or involve conditional distributions, which makes them difficult to apply. In the present paper we prove central limit theorems for sequences of dependent random variables of a certain special type which occurs frequently in mathematical statistics. The hypotheses do not involve conditional distributions.