91Ó°ÊÓ

A Markov chain is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. The problem gives us the following information: - The Markov chain has M states. - State J can be reached from state i. The question asks us to prove that state J can be reached from state i in M steps or less.

Let P(s) denote the probability of transitioning from a state i to state J. The transition probability matrix is defined as: \[ P = \begin{pmatrix} P_{1,1} & P_{1,2} & \cdots & P_{1,M} \\ P_{2,1} & P_{2,2} & \cdots & P_{2,M} \\ \vdots & \vdots & \ddots & \vdots \\ P_{M,1} & P_{M,2} & \cdots & P_{M,M} \\ \end{pmatrix}\] And we have: - \(P(s_j|s_i) = P_{i,j}\) - \(P(s_{j'}|s_j) \) for any eligible j' and j, where \(1 \leq j,j' \leq M\)

Let's consider the transition probability matrix \(P^m\) after m steps. The element in the i-th row and j-th column of \(P^m\), denoted as \(P_{i,j}^{(m)}\), represents the probability of going from state i to state J in exactly m steps. Since the total number of states is M, and we know that state J can be reached from state i, there must exist a sequence of transitions between states such that state J is reached eventually. It is possible that state J can be reached directly from state i in a single step, or through intermediate states. Our task is to prove that there exists a sequence of transitions such that state J can be reached in M steps or less. We can use the following approach: 1. If state J can be reached directly from state i in a single step, then we have proved the statement. 2. If not, let's assume that the longest path reachable from state i that doesn't include state J is of length L, with \(L < M\). Then let's try to make a transition from the L-th state in the chain to another state. There are two possibilities: a. There is a transition from the L-th state directly to state J, meaning that state J can be reached in L+1 steps or less. b. There is a transition from the L-th state to another state which is not J, but it will eventually lead to state J after K steps, with \(K \leq M - L\). This means that state J can be reached in L + K steps or less, and since K is at most M - L, we have that state J can be reached in M steps or less. In either of cases 2(a) or 2(b), we have shown that state J can be reached from state i in M steps or less, which completes the proof.