Let $\{X_t\}_{t\in\mathbb{Z}}$ be a stationary and ergodic stochastic process with finite second moment. Von Neuman's ergodic theorem implies that the time average $(1/N)\sum_{j=0}^{N-1} X_j$ converges to $\mathbb{E}X_0$ in $L^2$. In many textbooks, this fact is usually proved in an abstract setting of a Hilbert space via contraction or unitary operator arguments.
However, for example, if we further assume that $\{X_t\}_{t\in\mathbb{Z}}$ is i.i.d., then we can prove the $L^2$ weak law in a more direct way by calculating $\mathbb{E}|(1/N)\sum_{j=0}^{N-1} X_j -\mathbb{E}X_0|^2 \leq (1/N)\mathbb{E}X_0^2 = o(1)$.
My question: Instead of imposing independence, if we assume that $\{X_t\}_{t\in\mathbb{Z}}$ is a stationary ergodic Markov process, is there any simplification of the proof of the $L^2$ ergodic theorem? In other words, can we utilize the Markovian property to give a direct proof of the $L^2$ weak law?