Let $(X_{n}, Y_{n})_{n\in\mathbb{Z}}$ be a hidden Markov Model where the $(Y_{n})_{n}$ are the observations and the $(X_{n})_{n}$ are the hidden states which only take values 0 or 1. Assume that the Markov chain of the hidden states is irreducible and reccurent. I want to show that: $$ \mathbf{\frac{1}{n}\sum_{i=1}^{+\infty}\mathbb{P}(X_{i} = 0\mid Y_{-\infty:+\infty}) }\xrightarrow[n\rightarrow +\infty]{a.e} \mathbf{\mathbb{E}[\mathbb{P}(X_{0} = 0\mid Y_{-\infty:+\infty})]} $$ I think I can use Birkhoff's ergodic theorem but I can't see how to apply it to this specific case. Note that the random variable $\mathbb{P}(X_{i} = 0\mid Y_{-\infty:+\infty})$ is well-defined as the limit of the martingale $(\mathbb{P}(X_{i} = 0\mid Y_{-m:m}))_{m\in\mathbb{N}}$ and the hidden markov model $(X_{n}, Y_{n})_{n\in\mathbb{Z}}$ is also well-defined for negative indices (it suffices to take the distribution of $X_{0}$ as the stationary distribution and to extend the process to negative indices using Kolmogorov's extension theorem).
