Suppose we have an $\mathbb{R}$-valued Markov process, generating trajectories $x(t),\ t\geq 0$. Suppose it is stationary and ergodic. Its auto-covariance (ACF) function is defined as $$R(\tau) = \mathbb{E} (x(t)x(t+\tau)) .$$ It is rather common to estimate $R(\tau)$ by looking at a different quantity, the "sample ACF" $$ R^*_T(\tau) = \frac{1}{T}\int_{0}^T x(t)x(t+\tau) dt .$$
Question: under which conditions one can prove $R_T^* \to R,\ T\to\infty$ in some reasonable topology? For AR(p) it is known to be true, but what about other processes, for example, Hawkes processes?