Stationary distributions

Definition

A stationary distribution is a probability distribution that remains unchanged as the system evolves over time in a Markov chain. It represents a long-term behavior where the probabilities of being in each state stabilize, indicating that once the system reaches this distribution, it will continue to exhibit this distribution at subsequent time steps. Understanding stationary distributions is crucial for analyzing the long-term predictions and behaviors of Markov chains.

5 Must Know Facts For Your Next Test

1. Stationary distributions are unique for irreducible and aperiodic Markov chains, meaning that there is only one stationary distribution that describes the long-term behavior of the chain.
2. To find a stationary distribution, one must solve the equation $$oldsymbol{ u} P = oldsymbol{ u}$$, where $$oldsymbol{ u}$$ is the stationary distribution vector and $$P$$ is the transition matrix.
3. If a Markov chain has transient states, they do not contribute to the stationary distribution as their probabilities eventually drop to zero over time.
4. In practice, stationary distributions are useful in various fields such as queueing theory, economics, and genetics to understand steady-state behaviors.
5. The concept of stationary distributions can be extended to continuous-time Markov chains, where they still provide insights into long-term trends despite differences in time representation.

Review Questions

• How do stationary distributions relate to the long-term behavior of a Markov chain?
  • Stationary distributions provide insight into the long-term behavior of a Markov chain by indicating the probabilities of being in each state once equilibrium is reached. When a Markov chain converges to its stationary distribution, it means that no matter what initial state was chosen, the probabilities will stabilize and remain constant over time. This characteristic is essential for predicting how systems behave after many transitions.
• What steps are involved in determining the stationary distribution for a given Markov chain?
  • To determine the stationary distribution for a given Markov chain, you first need to identify its transition matrix. Then, you set up and solve the equation $$oldsymbol{ u} P = oldsymbol{ u}$$, along with the constraint that the sum of all probabilities in $$oldsymbol{ u}$$ equals 1. This process often involves linear algebra techniques to find the solution that represents the stationary distribution accurately.
• Evaluate how ergodicity affects the existence and uniqueness of stationary distributions in Markov chains.
  • Ergodicity plays a significant role in determining both the existence and uniqueness of stationary distributions in Markov chains. If a Markov chain is ergodic, it implies that it will eventually reach a unique stationary distribution regardless of its initial state due to its irreducibility and aperiodicity. This means that as time progresses, every state will be visited often enough for the probabilities to converge to this unique distribution, making it crucial for reliable long-term predictions.

Related terms

Markov chain: A Markov chain is a stochastic process that undergoes transitions between a finite or countable number of states, where the probability of transitioning to any particular state depends only on the current state.

Transition matrix: The transition matrix is a square matrix used to describe the probabilities of moving from one state to another in a Markov chain.

Ergodicity:

Ergodicity refers to a property of a Markov chain where, regardless of the starting state, the long-term behavior converges to a unique stationary distribution.