Stationary distribution

Definition

A stationary distribution is a probability distribution that remains unchanged as the system evolves over time in a Markov chain. This distribution reflects the long-term behavior of the chain, indicating the probabilities of being in each state after many transitions. When the Markov chain reaches its stationary distribution, the probabilities of being in each state become stable and no longer change with further iterations.

5 Must Know Facts For Your Next Test

1. The stationary distribution is unique for irreducible and aperiodic Markov chains, meaning there is only one distribution to which the chain converges over time.
2. To find the stationary distribution, you typically solve a system of linear equations derived from the transition matrix.
3. If a Markov chain is not ergodic, it may have multiple stationary distributions corresponding to different communicating classes.
4. The stationary distribution provides insight into the long-term behavior and steady-state probabilities of the system modeled by the Markov chain.
5. In practice, the concept of stationary distribution is used in various fields such as queueing theory, genetics, and economics to model systems that reach equilibrium.

Review Questions

• How does a stationary distribution relate to the concept of Markov chains and their long-term behavior?
  • A stationary distribution is fundamentally tied to Markov chains as it represents the probabilities associated with each state when the chain reaches equilibrium after many transitions. This means that regardless of the initial state, after enough time has passed, the system's behavior stabilizes according to this distribution. Understanding how this distribution emerges helps analyze how systems evolve over time and predict their eventual outcomes.
• What conditions must a Markov chain satisfy for it to have a unique stationary distribution, and why are these conditions important?
  • For a Markov chain to have a unique stationary distribution, it must be irreducible (all states can be reached from any other state) and aperiodic (the system does not cycle through states at fixed intervals). These conditions are crucial because they ensure that no matter where you start in the state space, you will converge to the same long-term behavior over time. This predictability is essential for modeling real-world systems where understanding equilibrium states is necessary.
• Evaluate how knowledge of stationary distributions can be applied in practical scenarios, such as queueing theory or economic modeling.
  • Knowledge of stationary distributions has significant applications in various fields. In queueing theory, understanding the stationary distribution helps in predicting average wait times and system performance under steady-state conditions. In economics, it can model consumer behavior patterns over time, providing insights into market equilibrium. By applying this concept effectively, analysts can make informed decisions based on long-term predictions rather than transient behaviors.

Related terms

Markov chain: A stochastic process that undergoes transitions from one state to another within a finite or countable number of possible states, where the probability of each transition depends only on the current state.

Ergodic: A property of a Markov chain where every state can be reached from any other state, ensuring that the chain will eventually converge to a unique stationary distribution.

Transition matrix: A matrix that describes the probabilities of moving from one state to another in a Markov chain, with each entry representing the probability of transitioning from a specific state to another.