To analyze this Markov chain, we’ll go through the following steps:

1. Define the State Space and Transition Matrix: The Markov chain has a state space $S = \{0, 1, 2\}$, and the one-step transition probability matrix is given as: $P = \begin{pmatrix} 0.75 & 0.25 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.75 & 0.25 \end{pmatrix}$

2. Initial Distribution: We are given that the initial distribution is uniform across states $\{0, 1, 2\}$, which means: $\pi_0 = \begin{pmatrix} \frac{1}{3} \\ \frac{1}{3} \\ \frac{1}{3} \end{pmatrix}$

3. Finding the Distribution After n Steps: To find the distribution after $n$ steps, $\pi_n$, we can use the relation: $\pi_n = \pi_0 \cdot P^n$ where $P^n$ is the $n$-step transition matrix, calculated by taking the power of the matrix $P$ for each step $n$.

4. Calculating $P^n$: Calculating $P^n$ by matrix exponentiation gives the probabilities of transitioning from one state to another in $n$ steps. For instance, for small values of $n$ (like $n = 1$ or $n = 2$), we can directly compute $P^n$ as follows:

   • For $n = 1$: $P^1 = P = \begin{pmatrix} 0.75 & 0.25 & 0 \\ 0.25 & 0.5 & 0.25 \\ 0 & 0.75 & 0.25 \end{pmatrix}$

   • For $n = 2$: $P^2 = P \cdot P = \begin{pmatrix} 0.625 & 0.3125 & 0.0625 \\ 0.3125 & 0.375 & 0.3125 \\ 0.0625 & 0.5625 & 0.375 \end{pmatrix}$

   This calculation can continue for any $n$ desired to find $\pi_n = \pi_0 \cdot P^n$.

5. Finding the Steady-State Distribution (Long-Term Behavior): For Markov chains, the steady-state distribution $\pi$ satisfies: $\pi = \pi \cdot P$ and $\sum_{i=0}^2 \pi_i = 1$.

   Solving the system of linear equations for $\pi$, we get: $\pi_0 = 0.2, \quad \pi_1 = 0.4, \quad \pi_2 = 0.4$ This is the steady-state distribution, which represents the long-term probability of finding the Markov chain in each state.

Would you like a step-by-step explanation of each calculation, or do you have questions about a particular step?

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Related Questions:

1. How would we interpret the steady-state distribution in practical terms for this Markov chain?
2. What changes if we modify one of the transition probabilities, say, changing $P(1,2)$ to 0.3?
3. Can we generalize this method to larger state spaces or more complex Markov chains?
4. How does the initial distribution affect the short-term and long-term behavior of the chain?
5. What if we are interested in the expected time to reach a specific state, such as state 2?

Tip: For Markov chains, studying both the $n$-step transition probabilities and the steady-state distribution provides insights into both short-term and long-term behavior.