A business executive is managing three branches, labeled A,...

A business executive is managing three branches, labeled $A, B$, and $C$, of a corporation. She never visits the same branch on consecutive days. If she visits branch $A$ one day, she visits branch $B$ the next day. If she visits either branch $B$ or $C$ that day, then the next day she is twice as likely to visit branch $A$ as to visit branch $B$ or $C$. Explain why the resulting transition matrix is regular. Which branch does she visit the most often in the long run?

Key Concepts

Regular Markov Chain

A regular Markov chain is characterized by the property that for some positive integer k, every entry of its kth power is positive. This means there is a nonzero probability of moving from any state to any other state in exactly k steps. This property guarantees that the chain is irreducible and aperiodic, which in turn ensures the existence and uniqueness of a stationary distribution and the convergence to it regardless of the starting state.

Transition Matrix

The transition matrix in a Markov chain framework contains the probabilities of moving from one state to another from one step to the next. Each row in this matrix must sum to one, reflecting the total probability from each state. The structure of the matrix directly incorporates any rules or restrictions of the system being modeled, such as the fixed transitions or relative probabilities described in the problem.

Stationary Distribution

The stationary distribution is a probability distribution over the states of a Markov chain that remains unchanged after application of the transition matrix. It represents the long-run proportion of time the chain spends in each state. This distribution gives insight into the steady-state behavior of the system, such as identifying which state (branch) is visited most frequently in the long run.

Limiting Behavior

The concept of limiting behavior in a Markov chain refers to the convergence of the state probabilities to the stationary distribution as time progresses. This behavior ensures that no matter what the initial state is, the probability distribution of the states will eventually stabilize, allowing predictions about long-term outcomes based solely on the stationary distribution.

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