Suppose $\{X(t)\}_{t>0}$ is a continuous time weakly stationary stochastic process, and suppose it has $0$ mean. Then $\gamma(t,h):=\text{Cov}[X(t),X(t+h)]=\mathbb{E}[X(t)X(t+h)]$ as a function is independent of $t$.
I have seen it stated that $$\gamma(h)=\lim_{T \to \infty}\frac{1}{T}\int_{0}^{T}X(t^{\prime})X(t^{\prime}+h)dt^{\prime}.$$
I suppose this is a continuous analogue of the law of large numbers, as because of stationarity we have that $X(t^{\prime})X(t^{\prime}+h)$ is identically distributed to $X(t^{\prime\prime})X(t^{\prime\prime}+h)$ for all $t^{\prime}\neq t^{\prime\prime}$. However, in the regular law of large numbers we also have independence, but I don't think it's the case that $X(t^{\prime})X(t^{\prime}+h)$ is independent of $X(t^{\prime\prime})X(t^{\prime\prime}+h)$ for all $t^{\prime} \neq t^{\prime\prime}$?
Can anyone point me to a reference for continuous time laws of large numbers applied to stationary processes?
Thank you for reading.