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When we talk about an ergodic Markov chain, we refer to a special type of Markov chain with some fascinating properties. The chain is said to be ergodic if it is possible to go from any state to any other state in some number of steps. This means that the Markov chain is irreducible, and it can explore the entire state space. It brings a sense of connectivity to the states.
Moreover, an ergodic Markov chain is also aperiodic, meaning that the time steps for returning to a state are not fixed multiples of a number greater than 1. Together, these properties ensure that the chain has a long-term behavior that reaches a steady state.

• A Markov chain is irreducible if it can move from any state to any other state.
• It is aperiodic if return times to a state are not fixed intervals.

This steady-state behavior is what allows us to identify a fixed probability vector, a key concept in understanding what happens as we let the Markov chain run in the long term.

The transition matrix, denoted as \(\mathbf{P}\), is a central concept in Markov chains. Imagine this matrix as a roadmap that guides how the chain moves from one state to another. Each entry \(p_{ij}\) in the matrix represents the probability of transitioning from state \(i\) to state \(j\). These probabilities are non-negative and each row sums up to 1, symbolizing a total probability of staying within some state.
The transition matrix of an ergodic Markov chain has some unique characteristics. Because of the irreducible nature of the chain, all row vectors in \(\mathbf{P}\) are eventually able to influence any state, ensuring that every state communicates with every other state.

• Each row in \(\mathbf{P}\) sums up to 1: \(\sum_j p_{ij} = 1\).
• \(p_{ij}\) reflects the chance of moving from state \(i\) to state \(j\).

Maintaining this structure helps in finding solutions like the fixed probability vector that remains the same under the chain's dynamics.

The concept of a fixed probability vector plays a pivotal role in understanding the behavior of Markov chains over the long term. Sometimes referred to as a steady-state vector, this is a vector \(\mathbf{w}\) that remains unchanged when the transition matrix \(\mathbf{P}\) is applied. Mathematically, it satisfies the equation \(\mathbf{w} \mathbf{P} = \mathbf{w}\).
This vector provides insight into the equilibrium distribution of the states, indicating what proportion of time the chain will spend in each state if allowed to run indefinitely.

• For \(\mathbf{w}\) to be valid, it must meet two criteria:
• The entries should be non-negative.
• Their sum must equal 1: \(\sum_i w_i = 1\).

By solving the equation \(\mathbf{w} \mathbf{P} = \mathbf{w}\), one deduces patterns in the chain's transitions, helping to predict long-term performance or results.

Detailed balance is a condition often used when working with Markov chains, especially when dealing with their reversibility. To satisfy detailed balance, there should exist a probability vector \(\mathbf{w}\) such that the transition probabilities are balanced in both directions between pairs of states. We can express it mathematically by saying \(w_i p_{ij} = w_j p_{ji}\).
This means for every pair of states \(i\) and \(j\), the weighted flow from \(i\) to \(j\) is the same as the flow from \(j\) back to \(i\). This symmetric relationship is crucial for reversible chains and helps in establishing the fixed probability vector for the reverse transition matrix, \(\mathbf{P}^{*}\).

• Reversibility: It implies that the chain looks the same if you reverse time.
• Guarantees that a fixed probability vector for \(\mathbf{P}\) is also valid for \(\mathbf{P}^{*}\).

By ensuring that detailed balance is maintained, one can confirm that both transition matrices share the same fixed probability vector, hence confirming the exercise statement.