The size of the averages of strongly mixing random variables

Mixing conditions have frequently been found to be satisfactory substitutes for independence in attempts to generalize the classical limit theorems of probability theory. We find that there is a natural way to classify stationary sequences with respect to the way in which the sizes of the corresponding averages are controlled by the moments of the marginal distribution, using two of the standard mixing conditions. Stationary sequences of [phi]-mixing random variables have averages controlled by their moments in the same way as independent sequences, whereas the weaker condition of strong mixing is shown, via examples, not to in general imply results analogous to the classical theorems.