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.