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The problem deals with a Markov chain \((X_n)_{n \geq 0}\) on a state space \(S = \{1, 2, \dots, m\}\) with a transition matrix \(P\). It involves collecting a reward based on a function \(f: S \to \mathbb{R}\), where a reward \(f(x)\) is received when the Markov chain is in state \(x\). The task is to prove a result regarding the expected total reward collected up to time \(n\).

### Step-by-step breakdown:

1. **Definition of the Markov Chain:**
   - The Markov chain is defined with transition matrix \(P\) and a finite state space \(S = \{1, 2, \dots, m\}\).
   - At each step \(n\), the Markov chain is in state \(X_n\), and a reward \(f(X_n)\) is collected.

2. **Reward Collection:**
   - The rewards are collected according to a function \(f(x)\), where \(x\) represents the current state of the chain.
   - The reward at each time step \(n\) is \(f(X_n)\).

3. **Probability Mass Function:**
   - For each \(n \geq 0\), \(r_n = [P(X_n = 1), \dots, P(X_n = m)]\) represents the probability mass function of \(X_n\) arranged as a row vector.

4. **Reward Vector \(v\):**
   - The reward function is represented by the column vector \(v = [f(1), f(2), \dots, f(m)]^T\), which lists the rewards corresponding to each state.

5. **Total Reward:**
   - The total reward collected up to time \(n\) is \(R_n = \sum_{k=0}^{n} f(X_k)\).

6. **Expected Total Reward:**
   - The goal is to prove the following formula for the expected total reward collected up to time \(n\):
   \[
   E[R_n] = r_0 (I + P + P^2 + \dots + P^n) v
   \]
   Here, \(r_0\) is the initial distribution (row vector), \(I\) is the identity matrix, and \(P^k\) is the transition matrix raised to the power \(k\).

### Outline of the Proof:

- The expected reward at time \(k\) is \(E[f(X_k)] = r_0 P^k v\).
- Thus, the expected total reward is the sum of the expected rewards from time 0 to \(n\):
  \[
  E[R_n] = \sum_{k=0}^{n} E[f(X_k)] = \sum_{k=0}^{n} r_0 P^k v
  \]
- Factor out \(r_0\) and \(v\):
  \[
  E[R_n] = r_0 \left( \sum_{k=0}^{n} P^k \right) v
  \]
- The summation \(\sum_{k=0}^{n} P^k\) can be written as \(I + P + P^2 + \dots + P^n\).
- Therefore, we have the desired result:
  \[
  E[R_n] = r_0 (I + P + P^2 + \dots + P^n) v
  \]

Would you like further details or explanations?

### Related Questions:
1. How is the expected value of the total reward derived from the Markov process?
2. What properties of the transition matrix \(P\) are used in this proof?
3. How would the proof change if the reward function \(f(x)\) were time-dependent?
4. Can this result be extended to an infinite time horizon?
5. How does the initial distribution \(r_0\) affect the expected total reward?

### Tip:
When dealing with Markov chains, understanding the power of the transition matrix is key, as it represents the probability of transitioning between states over multiple steps.

Let (X_n)_{n≥0} be a Markov chain on a state space S = {1, 2, …, m} with transition matrix P. Let f: S → R be a function. Suppose that we collect a reward f(x) when the Markov chain is at state x. For a given n ≥ 0, let r_n = [P(X_n = 1), …, P(X_n = m)] be the probability mass function of X_n arranged as a row vector. Let v = [f(1), f(2), …, f(m)]^T be the column vector representing the reward function f. Denote the total reward collected up to time n by R_n = Σ^n_{k=0} f(X_k). Prove that we have E[R_n] = r_0(I + P + P^2 + … + P^n)v.

Explore the mathematical proof and formula for calculating the expected reward in a Markov chain process. This problem delves into the probability mass function, transition matrices, and expected value calculations over time, designed for advanced students studying stochastic processes.

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