Stationary distribution

Definition

A stationary distribution is a probability distribution over the states of a stochastic process that remains unchanged as time progresses. It characterizes the long-term behavior of the system, showing the proportion of time the process spends in each state when it reaches equilibrium. This concept is essential for understanding Markov chains and regenerative processes, as it allows for the analysis of the system's stability and long-term trends.

5 Must Know Facts For Your Next Test

1. A stationary distribution exists if a Markov chain is irreducible and aperiodic, ensuring all states can be reached from any other state over time.
2. The entries of a stationary distribution sum to 1, representing total probability across all possible states in the system.
3. Once a Markov chain reaches its stationary distribution, the probabilities of being in each state remain constant over time.
4. For regenerative processes, stationary distributions help in computing important quantities like expected time until absorption or hitting certain states.
5. In many applications, finding the stationary distribution is crucial for predicting long-term outcomes in systems modeled by Markov chains.

Review Questions

• How does a stationary distribution relate to the behavior of a Markov chain over time?
  • A stationary distribution describes the long-term behavior of a Markov chain by indicating the proportion of time spent in each state when the chain has stabilized. As time progresses, if a Markov chain reaches its stationary distribution, the probabilities of being in different states no longer change. This means that regardless of where you start, after enough transitions, you will find yourself distributed according to this stationary distribution.
• In what situations would you expect to find a stationary distribution in regenerative processes, and why is it important?
  • You would expect to find a stationary distribution in regenerative processes that are both irreducible and aperiodic. This importance lies in its ability to provide insights into steady-state behavior and long-term performance measures. For instance, in risk assessment or queuing models, knowing the stationary distribution helps in understanding how systems behave over an extended period and can inform decision-making regarding resource allocation or service efficiency.
• Evaluate how changes in transition probabilities affect the stationary distribution and discuss potential implications for modeling real-world systems.
  • Changes in transition probabilities can significantly impact the stationary distribution by altering the likelihood of being found in certain states. If transition probabilities are increased for specific paths, the stationary distribution may shift to reflect this change, resulting in higher expected times in favored states. In real-world modeling—such as predicting customer behavior or system performance—this means that any adjustments to policies or conditions can lead to different long-term outcomes, emphasizing the need for continuous evaluation and adaptation of models to ensure accurate predictions.

Related terms

Markov chain:

A stochastic process that satisfies the Markov property, meaning that the future state depends only on the current state and not on the sequence of events that preceded it.

Ergodicity:

A property of a stochastic process where time averages equal ensemble averages, ensuring that the stationary distribution can be reached regardless of the starting state.

Transition matrix: A square matrix used to describe the probabilities of transitioning from one state to another in a Markov chain.