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 $q_1=\frac{\mu_2}{\mu_1}$?


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!

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.