Must eventual absorption be unlikely?

No. A quasi-stationary profile can exist even when eventual absorption occurs with probability one.

Is the distribution always unique?

No. Uniqueness depends on the process and assumptions.

Can machine learning help compute it?

Research methods use learning-based approaches, but finite-state cases can also be treated with established eigenvector or simulation methods; learning is not part of the definition.

AI Glossary / Mathematics and optimization / Advanced

Quasi-stationary distribution

Definition

A quasi-stationary distribution is a probability distribution over a process's nonabsorbing states that remains unchanged when the process starts with that distribution and is conditioned on not yet being absorbed. Absorption means reaching a state or set from which the modeled process does not return. The distribution describes the surviving population of trajectories, even while the probability of survival itself may keep falling.

What it means

Consider a system that moves among operating conditions before eventually failing permanently. Its ordinary long-run distribution may concentrate entirely on failure. That tells little about the relative operating conditions seen among systems that have not failed yet. A quasi-stationary distribution answers that conditional question. It does not imply that failure has stopped occurring or that individual surviving paths are static. The same idea appears in population models before extinction and stochastic processes before they leave a specified domain. Here the focus is the standard absorbing-process meaning of the term.

How it works

For a finite discrete-time Markov chain, separate the transition matrix among nonabsorbing states into a matrix Q. Its row sums can be below one because missing probability goes to absorption. A probability row vector q is quasi-stationary when qQ = lambda q for a survival factor lambda, with its entries nonnegative and summing to one. Dividing qQ by its total mass recovers q. Under suitable irreducibility and aperiodicity assumptions, distributions conditioned on survival can converge to this form from other starting states. Existence, uniqueness and convergence need their own conditions, particularly for infinite or reducible state spaces. Continuous-time models use a related generator formulation rather than the same one-step transition equation.

Three practical examples

An ecological model can describe relative population sizes among simulated populations that have not yet become extinct.
A reliability study can examine operating modes among devices that remain active before a modeled terminal failure.
A stochastic simulation can study states occupied before a process exits a designated region, distinguishing that conditional profile from overall exit risk. These uses require a credible transition model and do not make survival-conditioned observations representative of the full original population.

Think of it this way

Think of repeatedly photographing only the runners still participating in a race. Their relative distribution across parts of the course could settle into a stable pattern even as more runners drop out. The analogy captures conditioning on survival. It does not imply that the remaining runners stay in one place or that a stable composition means the number remaining is stable.

Worked example

Use three states: A, B and an absorbing state D. From A, move to A with probability 0.6, to B with 0.2 and to D with 0.2. From B, move to A with 0.1, to B with 0.7 and to D with 0.2. Once in D, remain there. The nonabsorbing matrix is Q = [[0.6, 0.2], [0.1, 0.7]]. Try q = (1/3, 2/3). Multiplication gives qQ = (0.6/3 + 0.2/3, 0.2/3 + 1.4/3) = (4/15, 8/15) = 0.8q. The total surviving mass after one step is 0.8. Renormalizing gives (1/3, 2/3) again, so q is quasi-stationary. After five steps, survival probability is 0.8^5 = 0.32768, while the surviving state's conditional proportions remain unchanged. The unconditional masses are approximately 0.10923 in A, 0.21845 in B and 0.67232 in D. Starting entirely in A would instead give conditional proportions (0.75, 0.25) after one step; quasi-stationarity is not an immediate property of every initial distribution.

Illustrative case study

In a fictional project, Maple Reliability models two active modes and a terminal shutdown state for a device class. Its dashboard initially shows only the mode proportions among operating devices, which appear stable. The analysts add survival probability and discover that stable conditional composition can coexist with a growing shutdown fraction. They compare the estimated transition model with observed transitions and explicitly state that repair is excluded from this absorbing model. The case illustrates interpretation without inventing a device failure rate or claiming that the small worked chain describes real equipment.

Common misconceptions and mistakes

Quasi-stationary does not mean approximately stationary in every sense. The conditioning is essential. It is not the ordinary stationary distribution of the full absorbing chain, and it is not automatically the time-average distribution along a path conditioned on a distant survival event. A conditional long-time limit, often called a Yaglom limit, is related to quasi-stationarity but its existence should not be presumed from the definition alone. Simply deleting the absorbing state and renormalizing every row creates a different Markov process; its stationary distribution need not equal the original process's quasi-stationary distribution when survival chances differ by state.

Business use

Use this concept when the question explicitly concerns behavior before a terminal event and ordinary stationarity hides that behavior. Define absorption carefully: permanent failure, extinction and leaving a study region are different events. Report the survival or absorption measure alongside the conditional state distribution. Check whether repairs, re-entry or policy changes make the absorbing model inappropriate. For estimated models, investigate uncertainty in transition probabilities and sensitivity to rare transitions. Avoid interpreting a healthy-looking survivor profile as evidence that the original population is healthy; conditioning changes which trajectories are represented.

Check your understanding

Conditional state proportions remain fixed while survival probability falls from 0.8 to 0.4. Is there a contradiction?

Answer: No. The proportions describe only surviving trajectories. Their total probability can shrink while their normalized composition stays the same.

Related terms

Stationary distribution
Conditional distribution
absorbing Markov chain
survival probability
Yaglom limit
substochastic matrix

Learn next

multiply the worked Q repeatedly from the initial state A, renormalizing by total surviving mass after each step. Compare the conditional proportions with q and track absorption separately. Then give A and B different absorption probabilities to explore how survival conditioning changes composition.

Initial state

Sources

Sources: Pablo A. Ferrari and Nevena Marić, Quasi stationary distributions and Fleming-Viot processes in countable spaces, introduction, https://arxiv.org/pdf/math/0605665 ; Zhiqiang Cai, Ling Lin and Xiang Zhou, Learn Quasi-stationary Distributions of Finite State Markov Chain, section 2, https://arxiv.org/pdf/2111.11213 . The three-state calculation and fictional reliability case are original illustrations.

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