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Chapter 4: Problem 38

Recall that state \(i\) is said to be positive recurrent if \(m_{i, i}<\infty\), where \(m_{i, i}\) is the expected number of transitions until the Markov chain, starting in state \(i\), makes a transition back into that state. Because \(\pi_{i}\), the long run proportion of time the Markov chain, starting in state \(i\), spends in state \(i\), satisfies $$ \pi_{i}=\frac{1}{m_{i, i}} $$ it follows that state \(i\) is positive recurrent if and only if \(\pi_{i}>0\). Suppose that state \(i\) is positive recurrent and that state \(i\) communicates with state \(j .\) Show that state \(j\) is also positive recurrent by arguing that there is an integer \(n\) such that $$ \pi_{j} \geqslant \pi_{i} P_{i, j}^{n}>0 $$

Short Answer

Expert verified
To show that state \(j\) is positive recurrent given state \(i\) communicates with state \(j\) and is positive recurrent, we find an integer \(n\) such that \(P_{i, j}^n > 0\). Then, we establish the inequality \(\pi_j \geq \pi_i P_{i,j}^n > 0\), which implies that \(\pi_j > 0\). Hence, state \(j\) is also positive recurrent.

Step by step solution

01

Recall that state \(i\) is said to be positive recurrent if \(m_{i, i} < \infty\), where \(m_{i, i}\) is the expected number of transitions until the Markov chain, starting in state \(i\), makes a transition back into that state. The long run proportion of time the Markov chain, starting in state \(i\), spends in state \(i\) is given by \(\pi_i = \frac{1}{m_{i, i}}\). Therefore, state \(i\) is positive recurrent if and only if \(\pi_i > 0\). Furthermore, we are given that state \(i\) communicates with state \(j\), which means there is some \(n\) such that \(P_{i, j}^n > 0\). #Step 2: Calculate \(\pi_j\)#

We need to show that \(\pi_j > 0\). Since \(i\) communicates with \(j\), there exists an integer \(n\) such that \(P_{i, j}^n > 0\). Therefore, we can write $$ \pi_j = \frac{1}{m_{j, j}} \geq \frac{\pi_i P_{i,j}^n}{1} $$ #Step 3: Establish the Inequality#
02

Now, we know that \(\pi_i > 0\) as state \(i\) is positive recurrent. Thus, we can multiply both sides of the inequality by \(\pi_i P_{i,j}^n\) and get $$ \pi_j \geq \pi_i P_{i,j}^n $$ Since \(\pi_i > 0\) and \(P_{i,j}^n > 0\), the product \(\pi_i P_{i,j}^n > 0\). Therefore, we have $$ \pi_j \geq \pi_i P_{i,j}^n > 0 $$ #Step 4: Conclude State \(j\) is Positive Recurrent#

The inequality \(\pi_j \geq \pi_i P_{i,j}^n > 0\) implies that \(\pi_j > 0\). Since \(\pi_j > 0\), state \(j\) is also positive recurrent.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Positive Recurrent
In the world of Markov chains, a state is considered positive recurrent if it provides a certain level of stability over long periods. Specifically, for a state \( i \), it is positive recurrent if the expected number of transitions, denoted by \( m_{i,i} \), for the chain to return to state \( i \) is finite. This expectation, \( m_{i,i} < \infty \), indicates that the system does not wander off indefinitely without coming back to \( i \). A handy way to check this property is by looking at the long-run proportion of time the chain spends in this state, represented by \( \pi_i = \frac{1}{m_{i,i}} \).
This means that if \( \pi_i > 0 \), state \( i \) is positive recurrent, assuring that the system revisits the state again and again steadily over time.
Long Run Proportion
A Markov chain can be in various states over time, but the long-run proportion tells us about the fraction of time it spends in any specific state as time tends to infinity. For a state \( i \), this proportion is denoted by \( \pi_i \). It's calculated as \( \pi_i = \frac{1}{m_{i,i}} \), where \( m_{i,i} \) is the average number of steps for the chain to return to state \( i \).
This formula captures a fundamental aspect: if a system revisits a state frequently, its long-run proportion is non-zero, and thus, if \( \pi_i > 0 \), the state is positive recurrent. This is not just a number; it's a powerful piece of information indicating how likely the system is to be found in state \( i \) over the long term.
Transition Probabilities
Transition probabilities are the building blocks of any Markov chain. These probabilities, \( P_{i,j}^{(n)} \), denote the likelihood that the chain moves from state \( i \) to state \( j \) in exactly \( n \) steps. These probabilities are crucial because they tell us about the dynamics of the system and how likely it is to switch states over time.
In the context of showing that a state \( j \) is positive recurrent, if state \( i \) is positive recurrent and communicates with \( j \), the transition probability from \( i \) to \( j \) over a certain number of steps \( n \) is non-zero (i.e., \( P_{i,j}^{n} > 0 \)). This indicates a tangible path exists from \( i \) to \( j \), reinforcing the concept that state \( j \) inherits the positive recurrence property from state \( i \).
State Communication
In Markov chains, states are said to "communicate" if it's possible to reach each from the other, albeit in possibly different numbers of steps. This mutual accessibility is a key feature because it ties together the behavior of two states.
In practical terms, if state \( i \) can reach state \( j \) with some non-zero probability \( P_{i,j}^{n} \) for some integer \( n \), then state \( i \) communicates with state \( j \). This has a significant implication: properties of recurrence or ergodicity can "flow" between communicating states. Thus, if \( i \) is found to be positive recurrent, and \( i \) communicates with \( j \), we can assert that \( j \) is also positive recurrent. This concept underlies the fact that states in the same communication class share the same long-term characteristics.

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Most popular questions from this chapter

For a time reversible Markov chain, argue that the rate at which transitions from \(i\) to \(j\) to \(k\) occur must equal the rate at which transitions from \(k\) to \(j\) to \(i\) occur.

Consider a process \(\left\\{X_{n}, n=0,1, \ldots\right\\}\) which takes on the values 0,1 , or 2 . Suppose $$ \begin{aligned} &P\left\\{X_{n+1}=j \mid X_{n}=i, X_{n-1}=i_{n-1}, \ldots, X_{0}=i_{0}\right\\} \\ &=\left\\{\begin{array}{ll} P_{i j}^{\mathrm{I}}, & \text { when } n \text { is even } \\ P_{i j}^{I I}, & \text { when } n \text { is odd } \end{array}\right. \end{aligned} $$ where \(\sum_{j=0}^{2} P_{i j}^{\mathrm{I}}=\sum_{j=0}^{2} P_{i j}^{\mathrm{II}}=1, i=0,1,2 .\) Is \(\left\\{X_{n}, n \geqslant 0\right\\}\) a Markov chain? If not, then show how, by enlarging the state space, we may transform it into a Markov chain.

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.

Show that if state \(i\) is recurrent and state \(i\) does not communicate with state \(j\), then \(P_{i j}=0 .\) This implies that once a process enters a recurrent class of states it can never leave that class. For this reason, a recurrent class is often referred to as a closed class.

Consider an irreducible finite Markov chain with states \(0,1, \ldots, N\). (a) Starting in state \(i\), what is the probability the process will ever visit state \(j\) ? Explain! (b) Let \(x_{i}=P\\{\) visit state \(N\) before state \(0 \mid\) start in \(i\\}\). Compute a set of linear equations which the \(x_{i}\) satisfy, \(i=0,1, \ldots, N\). (c) If \(\sum_{j} j P_{i j}=i\) for \(i=1, \ldots, N-1\), show that \(x_{i}=i / N\) is a solution to the equations in part (b)
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