A new covariance inequality and applications

We compare three dependence coefficients expressed in terms of conditional expectations, and we study their behaviour in various situations. Next, we give a new covariance inequality involving the weakest of those coefficients, and we compare this bound to that obtained by Rio (Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 587–597) in the strongly mixing case. This new inequality is used to derive sharp limit theorems, such as Donsker's invariance principle and Marcinkiewicz's strong law. As a consequence of a Burkhölder-type inequality, we obtain a deviation inequality for partial sums.