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Chapter 4: Problem 15

Prove that if the number of states in a Markov chain is \(M\), and if state \(j\) can be reached from state \(i\), then it can be reached in \(M\) steps or less.

Short Answer

Expert verified
In a Markov chain with $M$ states, if state $j$ can be reached from state $i$, there exists an integer $n \le M$ such that the probability of transitioning from state $i$ to state $j$ in exactly $n$ steps, \(P_n(i, j)\), is greater than zero. This is because the longest possible path, represented by a directed graph with nodes for states and edges for transitions, cannot exceed $M-1$ steps without visiting a state more than once. Removing any loop with a revisit results in a shorter path with non-zero probability, proving that state $j$ can be reached in $M$ steps or less.

Step by step solution

01

Define the Markov chain and its properties

A Markov chain is a stochastic process with the Markov property, meaning that the future state is dependent only on the current state and not on the previous states. Formally, let \(X = (X_0, X_1, \dots , X_n)\) represent the sequence of states of a discrete-time Markov chain with a finite state space \(M\). Then, a Markov chain satisfies the property \(P(X_{n+1} = x_{n+1} | X_n = x_n, X_{n-1} = x_{n-1}, \dots , X_0 = x_0)=P(X_{n+1}=x_{n+1}|X_n = x_n)\), for all \(x_0, x_1, \ldots, x_n, x_{n+1}\) and all \(n \ge 0\).
02

Reachability in Markov chains

In Markov chains, we say that state \(j\) is reachable from state \(i\) if there exists a sequence of steps such that the Markov chain can transition from state \(i\) to state \(j\). Mathematically, state \(j\) is reachable from state \(i\) if there exists an integer \(n \ge 1\) such that the probability of transitioning from state \(i\) to state \(j\) in exactly \(n\) steps, \(P_n(i, j)\), is greater than zero.
03

Formulating the proof

To prove that state \(j\) can be reached from state \(i\) in \(M\) steps or less, we first assume that state \(j\) is reachable from state \(i\). We then need to show that there exists an integer \(n \le M\) such that the probability of transitioning from state \(i\) to state \(j\) in exactly \(n\) steps, \(P_n(i, j)\), is greater than zero.
04

Proving the statement

Since state \(j\) can be reached from state \(i\), there exists a sequence of steps of length at most \(M-1\), such that the probability of transitioning from state \(i\) to state \(j\) is greater than zero. We can imagine a directed graph with nodes representing the states and edges signifying possible transitions between these states, where the edge weights represent the transition probabilities. Let's consider the longest path from state \(i\) to state \(j\) in this graph. Since we have \(M\) states, this path cannot have a length greater than \(M-1\) without visiting at least one state more than once. If there's a state that is visited more than once in the path, we can remove the loop composed of the extra visit and obtain a shorter path with non-zero probability, as the transition probabilities are non-negative. Therefore, there exists a path of length at most \(M-1\) between state \(i\) and state \(j\), which means that it can be reached in \(M\) steps or less.

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Most popular questions from this chapter

An individual possesses \(r\) umbrellas that he employs in going from his home to office, and vice versa. If he is at home (the office) at the beginning (end) of a day and it is raining, then he will take an umbrella with him to the office (home), provided there is one to be taken. If it is not raining, then he never takes an umbrella. Assume that, independent of the past, it rains at the beginning (end) of a day with probability \(p\). (a) Define a Markov chain with \(r+1\) states, which will help us to determine the proportion of time that our man gets wet. (Note: He gets wet if it is raining, and all umbrellas are at his other location.) (b) Show that the limiting probabilities are given by $$ \pi_{i}=\left\\{\begin{array}{ll} \frac{q}{r+q}, & \text { if } i=0 \\ \frac{1}{r+q}, & \text { if } i=1, \ldots, r \end{array} \quad \text { where } q=1-p\right. $$ (c) What fraction of time does our man get wet? (d) When \(r=3\), what value of \(p\) maximizes the fraction of time he gets wet

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