My question concerns the adjustment of Birkhoff's ergodic theorem for Markov processes to the case of quasi-stationary distributions. I try to recall that theorem to the best of my capabilities and then state my use-case and then the question.
The ergodic theorem for stochastic processes $x(t)\in E$ on some "nice space" (lets say Polish, but in my case a finite discrete space) looks at the limiting behaviour of the time average of $x(t)$ over an interval $T$ as $T\to\infty$. I'm interested in the Markov case for $E$ discrete, but $t$ continuous. For a time continuous homogeneous Markov processes on $[0,\infty)$ with transition probability $P$ and initial distribution $\rho$ one may construct a Markov measure $\mathbb P_\rho$ (somehow compatible with the Lebesgue measure I guess) on the space $E^{[0,\infty)}$, where the trajectories live. Then Birkhoff's ergodic theorem takes a form along the following lines: Let $\rho^\star$ be a stationary distribution with respect to $P$, then \begin{equation} \mathcal Q(T)=\frac{1}{T}\int_{t=0}^Tg\Big(x(s)\Big)\lambda(ds) \end{equation} converges for $\mathbb P_{\rho^\star}$-almost-all trajectories $x(t)$ and any function $g:E\to \mathbb R$ (no integrability conditions on $g$ since $E$ is discrete) to the expectation value $\rho(g)$ or a version thereof (i.e. for almost all times). I am interpreting freely from these notes here. Please correct me, if it's incorrect.
In my case I have not a stationary distribution, but a quasi-stationary one. The application is an $SIS$-compartmental epidemiological model on a graph. The state-space $E=\{S,I\}^N$ is discrete and the transition rates can be deduced from the recovery and infection rates of the individual nodes. The state where all nodes are susceptible is absorbing and the distribution that has support only on that state is stationary for any set of parameters so long as the recovery rate is positive. There are also parameter regimes where the process also has a quasi-stationary distribution, in the sense that eventually it will reach the stationary disease-free distribution but if that disease-free state is extracted from the state-space together with the transition rates (?), then the distribution would be stationary. Unfortunately the reference is behind paywalls, so I cannot find the actual definition and instead recite the wikipedia reference, which in my reading is a bit vague on what the Markov process would look like if one state was extracted.
Question: I am wondering whether there are statements about the behaviour of $\mathcal Q(T)$ for quasi-stationary distributions $\rho$ and, say, finite $T$. Something to the extend that it will "mostly" be fairly "close" to $\rho(g)$?