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In the paper, Survival in a quasi-death process by van Doorn and Pollett, the quasi-stationary distribution of a transient CTMC is discussed and QSD for a simple death process is derived.

Consider a CTMC with an absorbing state; let $Q$ be its infinitesimal generator and $Q_a$ be the reduced generator after the removal of the absorbing state. As long as $Q_a$ is irreducible, it is well-known in literature that a probability vector $q$ satisfying $q^\top Q_a=\theta q$ gives the limiting conditional distribution and QSD.

van Doorn and Pollett consider the case when $Q_a$ is reducible. They state and prove the conditions for the existence of a QSD measure in such a case. My question is on how to interpret one of their results for a simple death process. Let the death process $X$ have states $\{2,1,0\}$; it starts from 2 and gets absorbed at 0. The generator is $\begin{bmatrix}-\mu_2 & \mu_2 & 0\\0 & -\mu_1 & \mu_1 \\ 0 & 0 & 0\end{bmatrix}$. If $P(X(0)=2)=1$ and $\mu_2<\mu_1$, the QSD is $q_1=\frac{\mu_2}{\mu_1}$ and $q_2=1-q_1$, which is obtained by solving an equation of the form $q^\top Q_a=\theta q$.

How does one make sense of these probabilities? I understand the QSD probability has a (Yaglom) limit interpretation: $q_i=\lim_{t\to \infty} P(X(t)=i\mid T>t)$, where $T$ is the time of absorption. In the case of a positive recurrent CTMC, the stationary probability $\pi_i$ is the limit of the fractional time one spends in state $i$. However in this case, the expected fractional time one spends in state 2, conditioned on non-absorption is $\frac{1/\mu_2}{1/\mu_1+1/\mu_2}$, which is different from $q_2$. What is a way to understand this metric? Were I to run a simple simulation, which quantity will equal $q_1=\frac{\mu_2}{\mu_1}$?

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First, I'm a bit confused as to how the matrix is set up, but it seems the rows and columns are laid out in the order $2,1,0$, is that correct?

And I think that the Yaglom limit is really the right way to interpret $q_i$. To say a bit more, let's say you do simulate many chains.

If you just look at the fractional time they spend in $i$ at a large time $t$ you will get something tending to $0$ (eventual absorption is certain).

What you are really interested in looking at is the fractional time that the chains which survived up until time $t$ spend in state $i$ at $t$. You are only concerned with the chains that survived, and of course this quantity is just the Yaglom limit.

Sorry if this appears too late!

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