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Chapter 4: Problem 38

Recall that state \(i\) is said to be positive recurrent if \(m_{i, i}<\infty\), where \(m_{i, 1}\) 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 $$ \dot{\pi}_{i}=\frac{1}{m_{i, i}} $$ 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
In conclusion, if state \(i\) is positive recurrent and communicates with state \(j\), then state \(j\) is also positive recurrent because it satisfies the condition \(\pi_j > 0\).

Step by step solution

01

Define the terms

Before starting the proof, let's define the terms given in the problem statement: 1. State \(i\) is said to be positive recurrent if \(m_{i,i}<\infty\), where \(m_{i,1}\) is the expected number of transitions until the Markov chain, starting in state \(i\), makes a transition back into that state. 2. \(\pi_i\) is the long run proportion of time the Markov chain, starting in state \(i\), spends in state \(i\), satisfies \(\dot{\pi}_{i}=\frac{1}{m_{i,i}}\). It follows that state \(i\) is positive recurrent if and only if \(\pi_i > 0\). 3. State \(i\) communicates with state \(j\) if there is a positive probability that the chain will eventually transition from state \(i\) to state \(j\) and vice versa, meaning that \(P_{i,j} > 0\) and \(P_{j,i} > 0\). Now, let's proceed with the proof to show that state \(j\) is also positive recurrent.
02

Use the relationship between communicating states

As state \(i\) communicates with state \(j\), there exists a positive integer \(n\) such that the probability of transitioning from state \(i\) to state \(j\) in \(n\) steps is greater than 0, i.e., \(P_{i,j}^n > 0\).
03

Utilize the properties of positive recurrent states

Since state \(i\) is positive recurrent, we have \(\pi_i > 0\).
04

Show that state j is positive recurrent

Now, we can write: $$\pi_j \geqslant \pi_i P_{i, j}^n$$ Since both \(\pi_i > 0\) and \(P_{i,j}^n > 0\), it follows that: $$\pi_{j} > 0$$ Hence, state \(j\) is also positive recurrent because it satisfies the condition \(\pi_j > 0\). In conclusion, if state \(i\) is positive recurrent and communicates with state \(j\), then 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.

Markov Chain Transitions
Understanding Markov chain transitions is crucial for analyzing how a system evolves over time. A Markov chain represents a stochastic process where the next state depends only on the current state, not on the sequence of events that preceded it.

To put it simply, imagine you're playing a board game where your next move is determined solely by your current position and a roll of the dice. This is akin to how Markov chains operate, with each roll being a transition and your position on the board representing a state. In mathematical terms, if we're at state i, the probability that we'll move to state j in the next step is denoted as Pi,j. These probabilities form a transition matrix that encapsulates all possible moves within the chain.

When it comes to real-world applications, we could be looking at things like predicting weather (will it be sunny, rainy, or cloudy tomorrow given today's weather?) or modeling queues in a customer service center (will the next customer wait, be served, or leave?). Each transition provides a piece of the puzzle that helps us understand the complex picture of how the system behaves over time.
Long Run Proportion of Time
The long run proportion of time a state is occupied in a Markov chain is a fundamental concept used to measure how often the system is in a particular state in the long term. This concept is closely related to the idea of steady-state probabilities, which are probabilities that the system settles into after a large number of transitions.

Consider watching TV and flipping through channels randomly. Over a very long period, you might notice that you spend a certain fraction of time on each channel, irrespective of your starting channel. This notion is what the long run proportion signifies in the context of Markov chains. Mathematically, it is represented by \( \pi_i \) for state i, and it is related to the expected number of steps to return to state i, denoted \( m_{i,i} \).

The relationship between these two measurements is pivotal since it tells us that if \( \pi_i > 0 \), then the state i is frequently visited, categorizing it as positive recurrent. In essence, \( \pi_i \) serves not only as a statistical measure but also as an indicator of the 'health' or 'stability' of that state within the system dynamics.
Communicating States
The concept of communicating states brings to light the interconnectedness within a Markov chain. Two states i and j are said to communicate with each other if it's possible to go from state i to state j and back, potentially after several transitions.

This is akin to being in a neighborhood where every house is reachable by walking from any other house, though the path might be indirect. In our mathematical neighborhood, if i and j communicate, it means there are positive probabilities \( P_{i,j} \) and \( P_{j,i} \) that you can move from one state to the other. This concept is crucial because it ensures that the Markov chain behaves cohesively and that every state influences and is influenced by others.

Moreover, communication between states implies a shared fate in terms of recurrence. If state i is positive recurrent and there's a path to state j, it means j will also be visited frequently. Therefore, j becomes positive recurrent as well. The ability for states to communicate guarantees that certain properties, such as recurrence, are preserved across the chain and allows for a unified analysis of the system's behavior over time.

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Most popular questions from this chapter

For a branching process, calculate \(\pi_{0}\) when (a) \(P_{0}=\frac{1}{47} \cdot P_{2}=\frac{3}{4}\) (b) \(P_{0}=\frac{1}{4}, P_{1}=\frac{1}{2}, P_{2}=\frac{1}{4}\) (c) \(P_{0}=\frac{1}{6}, P_{1}=\frac{1}{2}, P_{3}=\frac{1}{3}\)

On a chessboard compute the expected number of plays it takes a knight, starting in one of the four corners of the chessboard, to return to its initial position if we assume that at each play it is equally likely to choose any of its legal moves. (No other pieces are on the board.) Hint: Make use of Example 4.32.

A transition probability matrix \(\mathbf{P}\) is said to be doubly stochastic if the sum over each column equals one; that is, $$ \sum_{i} P_{i j}=1, \quad \text { for all } j $$ If such a chain is irreducible and aperiodic and consists of \(M+1\) states \(0,1, \ldots, M\), show that the limiting probabilities are given by $$ \pi_{j}=\frac{1}{M+1}, \quad j=0,1, \ldots, M $$

\(M\) balls are initially distributed among \(m\) urns. At each stage one of the balls is selected at random, taken from whichever urn it is in, and then placed, at random, in one of the other \(M-1\) urns. Consider the Markov chain whose state at any time is the vector \(\left(n_{1}, \ldots, n_{m}\right)\) where \(n_{i}\) denotes the number of balls in urn \(i\). Guess at the limiting probabilities for this Markov chain and then verify your guess and show at the same time that the Markov chain is time reversible.

Each morning an individual leaves his house and goes for a run. He is equally likely to leave either from his front or back door. Upon leaving the house, he chooses a pair of running shoes (or goes running barefoot if there are no shoes at the door from which he departed). On his return he is equally likely to enter, and leave his running shoes, either by the front or back door. If he owns a total of \(k\) pairs of running shoes, what proportion of the time does he run barefooted?
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