Let $X(t)$ be a random process. It is well known that stationarity is more strict condition than wide-sense stationarity. In a stationary random process, the pdf of a random variable $X(t_i)$ is the same at any time instance $t_i$. On the other hand, in a wide-sense stationary process, every random variable $X(t_i)$ has the same mean and variance, while the pdf's may be different for different time instances $t_i$. In other words, we can say that the wide-sense stationary random process has the time-invariant ensemble average and variance.
The autocorrelation function depends just on the time difference $\tau$, i.e. $R(\tau)$, no matter if we talk about the stationary of the wide-sense stationary process.
In an ergodic random process, the time-invariant ensemble average must be equal (with probability one) to the time average of every single realization of the random process $X(t)$.
Now assume that we have an ergodic random process $X(t)$. Is $X(t)$ stationary or wide-sense stationary then? I read somewhere that ergodicity implies stationarity, but I think that ergodicity implies just wide-sense stationarity. The reason follows from the definition of ergodic random process. That is, if we have an ergodic random process, we know that it has the time-invariant ensemble average which is the criteria set for the wide-sense stationary process.
I would like someone to correct me if I am wrong, or to confirm it.