Signal Plus Weakly Stationary Noise
For ergodic processes, time averages (defined by integrals over time) and ensemble averages (defined by expectations with respect to probability distributions) are identical. This means that the autocovariance is the same, no matter if defined by a time integral or by an expectation:

$$C_X(\tau)=\lim_{T\rightarrow\infty}\frac{1}{2T}\int_{-T}^{T}X(t+\tau)X(t)dt=
E\{X(t+\tau)X(t)\}$$