Two necessary and sufficient conditions of mean ergodicity about covariance stationary process

Mean ergodicity is an important property of covariance stationary process and has extensive application in practice. If a covariance stationary process obeys a mean ergodic theorem, we can estimate the mean value of covariance stationary process affectivity. We present two equivalent definitions of the mean ergodicity about covariance stationary process, and obtain two sufficient and necessary conditions of mean ergodicity and a simple corollary for compound covariance stationary process. We also point out the similar results for real covariance stationary process. For illustration, we present two numerical examples.