A Markov chain that has a limiting distribution is described as which type?

A Markov chain that has a limiting distribution is described as which type?

• A. Ergodic chain
• B. Aperiodic chain
• C. Irreducible chain
• D. Positive recurrent chain

Answer: (B)

A limiting distribution occurs when a Markov chain is ergodic, meaning it is irreducible and aperiodic. Irreducible ensures every state can be reached from every other state, so the chain has a single long-run behavior. Aperiodicity ensures the chain doesn’t get trapped in fixed cycles, allowing the distribution to settle down. Together, these properties guarantee that the n-step transition matrix P^n converges to a matrix with identical rows equal to the stationary distribution pi, which satisfies pi = pi P. If the chain isn’t irreducible, you could end up with different long-run behaviors in different parts of the state space, so there wouldn’t be a single limiting distribution. If it’s irreducible but still periodic, convergence may fail even though a stationary distribution exists. Thus, the proper description is an ergodic chain.

A limiting distribution occurs when a Markov chain is ergodic, meaning it is irreducible and aperiodic. Irreducible ensures every state can be reached from every other state, so the chain has a single long-run behavior. Aperiodicity ensures the chain doesn’t get trapped in fixed cycles, allowing the distribution to settle down. Together, these properties guarantee that the n-step transition matrix P^n converges to a matrix with identical rows equal to the stationary distribution pi, which satisfies pi = pi P. If the chain isn’t irreducible, you could end up with different long-run behaviors in different parts of the state space, so there wouldn’t be a single limiting distribution. If it’s irreducible but still periodic, convergence may fail even though a stationary distribution exists. Thus, the proper description is an ergodic chain.