For each realization, both the time-averaged mean and the time-averaged autocorrelation function converge to the corresponding ensemble averages. Thus the random-phased sinusoid is ergodic in both mean and autocorrelation.

Remark

A random process is ergodic if its ensemble averages converge in the m.s. sense to the corresponding time averages. This is a stronger requirement than stationarity- the ensemble averages of all orders of such a process are independent of time. This implies that an ergodic process is necessarily stationary in the strict sense. The converse is not true- there are stationary random processes which are not ergodic.