In this note we establish an inequality between the maximal coefficient of correlation and the φ -mixing coefficient which is symmetric in its arguments. Motivated by this inequality, we introduce a mixing coefficient which is the product of two φ -mixing coefficients.
We also study an invariance principle under conditions imposed on this new mixing coefficient. As a consequence of this result it follows that the invariance principle holds when either the direct-time process or its time-reversed process is φ -mixing; when both processes are φ-mixing the invariance principle holds for sequences of L2-integrable random variables under a mixing rate weaker than that used by Ibragimov.
