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In a regular Markov chain, the transition matrix plays a critical role in determining how the system progresses from one state to another. A transition matrix, often denoted by \( \mathbf{P} \), is a square matrix where each entry \( P_{ij} \) represents the probability of transitioning from state \( i \) to state \( j \). This matrix is fundamental to analyzing Markov chains.

To qualify as a regular transition matrix, a specific power of this matrix must have all positive entries. This means that there is a positive probability of reaching any state from any other state in the chain, no matter where you start. Such a quality ensures that the Markov chain is ergodic, implying the chain is irreducible and aperiodic. This condition gives rise to a unique stationary distribution, satisfying certain stable properties as time progresses.

The transition matrix is essential, as it not only describes the immediate behavior of the chain but also determines critical long-term properties, like the stationary distribution.

The fixed probability vector, more formally known as the stationary distribution, is a key concept in Markov chains and is necessary for understanding their long-term behavior. Represented as \( \mathbf{w} \), it is a row vector that remains unchanged when multiplied by the transition matrix \( \mathbf{P} \). Mathematically, it satisfies the equation: \( \mathbf{w} \mathbf{P} = \mathbf{w} \).

This means that once the chain reaches this distribution, the probabilities of being in each state no longer change as steps proceed. The stationary distribution represents a kind of equilibrium where the relative proportions of time the system spends in each state become constant.

For a regular Markov chain, this stationary distribution is unique, which is a direct consequence of the matrix \( \mathbf{P} \) being regular. The repeated rows of this vector form the matrix \( \mathbf{W} \), which when multiplied by \( \mathbf{P} \) gives back \( \mathbf{W} \), demonstrating that \( \mathbf{W} \) captures the enduring nature of the system.

The stationary distribution of a Markov chain is essentially what gives the chain its long-term characteristics. It tells us the proportion of time that the chain will spend in each state in the long run. This distribution is crucial for understanding how a Markov chain behaves over an extended period. All states in the chain communicate with each other in a regular chain, ensuring its uniqueness.

When represented as a matrix \( \mathbf{W} \), composed of identical rows of the stationary distribution \( \mathbf{w} \), it has an interesting property: multiplying \( \mathbf{W} \) by itself repeatedly (i.e., \( \mathbf{W}^{k} \) for any positive integer \( k \)) still results in \( \mathbf{W} \). This equality signifies that the equilibrium proportions remain unchanged through the progression.

Understanding stationary distributions helps in various practical scenarios, like predicting steady states or equilibrium in systems modeled by regular Markov chains. It gives insights into the likelihood of finding the system in any particular state, thereby serving as a powerful tool in fields like stochastic processes, queuing theory, and economics.