An inequality involving correlations

A useful inequality involving correlations between three random variables with mean zero and finite second moments is presented. The inequality is applied to show that the entries on the main diagonal of the spectral density of a periodically correlated Markov process, as derived in Nematollahi and Soltani [2000. Discrete time periodically correlated Markov processes. Probab. Math. Statist. 20 (1) 127–140], are nonnegative. The inequality, when two variables with the same second moments are involved, is compared to the Cauchy–Schwartz inequality.