Quasistationary Probability Distribution

Definition

In birth and death processes where the origin is absorbing (certain extinction), there is no stationary distribution. However, prior to extinction, the probability distribution of $X(t)$ can be approximately stationary for a long period of time (especially if the extinction time is very long).

This approximate stationary distribution is called the quasistationary probability distribution (or quasiequilibrium).

Definition

Conditioned on non-extinction:

$q_i(t)=\frac{p_i(t)}{1-p_0(t)},i=1,2,\dots$

The quasistationary distribution is the limit: $q_i^{\ast }={\lim }_{t\to \infty }{q}_i(t)$.

Interpretation

If you observe the process and it hasn’t gone extinct for a very long time, $q_i^{\ast }$ gives the probability that the population is currently at size $i$. It describes the “metastable” behavior before eventual extinction.

Formula for Finite State Space $\{0,1,\dots ,N\}$

$q_i^{\ast }=\frac{\frac{{\lambda }_1{\lambda }_2\cdots {\lambda }_{i-1}}{{\mu }_1{\mu }_2\cdots {\mu }_i}}{\sum _{j=1}^N\frac{{\lambda }_1{\lambda }_2\cdots {\lambda }_{j-1}}{{\mu }_1{\mu }_2\cdots {\mu }_j}},i=1,\dots ,N$

where ${\lambda }_1\cdots {\lambda }_0/{\mu }_1\cdots {\mu }_1\equiv 1/{\mu }_1$ when $i=1$.

Matrix Form

Let $D$ be the $N\times N$ submatrix of the generator $Q$ (delete first row and column). The quasistationary distribution satisfies:

$D\mathbf{q}=0,{\sum }_{i=1}^N{q}_i=1$

Comparison with Stationary Distribution

	Quasistationary ($q^{\ast }$)	Stationary ($\pi$)
Origin	$q_0^{\ast }=0$ (conditioned away)	${\pi }_0>0$
Existence	Always exists for finite $N$	Requires $\sum {\theta }_i<\infty$
Interpretation	Conditional on survival	Long-run proportion

Related

• Probability of Population Extinction
• Expected Time to Extinction
• Stochastic Logistic Growth Process
• Stochastic SIS Epidemic Model