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Chapter 4: Problem 13

Let \(\mathbf{P}\) be the transition probability matrix of a Markov chain. Argue that if for some positive integer \(r, \mathbf{P}^{r}\) has all positive entries, then so does \(\mathbf{P}^{n}\), for all integers \(n \geqslant r\).

Short Answer

Expert verified
We can write any integer \(n \geqslant r\) as \(n = qr + s\), where \(q\) and \(s\) are integers, \(q \geq 1\), and \(0 \leq s < r\). If \(\mathbf{P}^r\) has all positive entries, then we can express \(\mathbf{P}^n = (\mathbf{P}^r)^q \mathbf{P}^s\), and since the product of matrices with all positive entries also has all positive entries, it follows that \(\mathbf{P}^n\) must have all positive entries for all integers \(n \geqslant r\).

Step by step solution

01

Write n as a multiple of r plus an additional positive integer

Consider an integer \(n \geqslant r\). Since r is a positive integer, we can always write n as $$ n = qr + s $$ where \(q\) and \(s\) are integers, \(q \geq 1\), and \(0 \leq s < r\).
02

Express \(\mathbf{P}^n\) using \(\mathbf{P}^r\) and \(\mathbf{P}^s\)

Now, let's use this representation of \(n\) to express the matrix \(\mathbf{P}^n\) in terms of \(\mathbf{P}^r\) and \(\mathbf{P}^s\): $$ \mathbf{P}^n = \mathbf{P}^{qr + s} = (\mathbf{P}^r)^q \mathbf{P}^s $$ Here, we used the property of matrix exponentiation which states that \(\mathbf{A}^{m+n} = \mathbf{A}^m \mathbf{A}^n\) for any matrix \(\mathbf{A}\) and positive integers \(m\) and \(n\).
03

Use positivity of \(\mathbf{P}^r\) to show positivity of \(\mathbf{P}^n\)

We know that \(\mathbf{P}^r\) has all positive entries. Therefore, \((\mathbf{P}^r)^q\) must also have all positive entries, regardless of the value of \(q\). Now, we can use the property that the product of two matrices with all positive entries will also have all positive entries. Since \((\mathbf{P}^r)^q\) has all positive entries, and \(\mathbf{P}^s\) is a transition probability matrix (which has non-negative entries), their product, \(\mathbf{P}^n\), must also have all positive entries. This proves that if \(\mathbf{P}^r\) has all positive entries, then \(\mathbf{P}^n\) will also have all positive entries for all integers \(n \geqslant r\).

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Key Concepts

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

Transition Probability Matrix
A transition probability matrix, often denoted as \( \mathbf{P} \), is a fundamental component of a Markov chain. This matrix is used to describe the probabilities of transitioning from one state to another in a stochastic process. Each element \( p_{ij} \) of the matrix represents the probability of moving from state \( i \) to state \( j \). The key properties of a transition probability matrix include:
  • Each row sums to 1, indicating that the total probability of moving from a given state to any other is 100%.
  • All entries are non-negative, as probabilities cannot be negative.
Understanding the transition probability matrix is crucial as it allows us to see patterns and probabilities in state transitions over time.
Matrix Exponentiation
Matrix exponentiation is a method used to compute the power of matrices, particularly transition probability matrices in Markov chains. In simpler terms, raising a matrix \( \mathbf{P} \) to a power \( n \), denoted \( \mathbf{P}^n \), involves multiplying the matrix by itself \( n \) times. This concept is essential when analyzing long-term behavior of Markov chains:
  • For a transition probability matrix \( \mathbf{P} \), the matrix \( \mathbf{P}^n \) gives the \( n \)-step transition probabilities.
  • Using matrix exponentiation helps predict the state of the system after multiple transitions, crucial for understanding steady-state behavior.
Matrix exponentiation provides insights into how probabilities evolve in a Markov chain and enables us to deduce characteristics like stability and convergence.
Positive Entries
When a matrix, like the transition probability matrix \( \mathbf{P}^r \) of a Markov chain, is said to have positive entries, it means every element in the matrix is greater than zero. This idea is significant because:
  • A matrix with all positive entries indicates that there is a non-zero probability of moving between any two states in \( r \) steps.
  • This property can influence the connectedness of the state space, suggesting that all states communicate with one another in the given steps.
In the context of the provided exercise, if \( \mathbf{P}^r \) has all positive entries, then \( \mathbf{P}^n \) will retain these positive entries for all \( n \geq r \), ensuring persistent transition possibilities among states.
Stochastic Process
A stochastic process is a sequence of random variables that represent a system evolving over time in probabilistic terms. In the case of Markov chains, the stochastic process involves states transitioning from one to another, with certain probabilities defined by the transition probability matrix. Stochastic processes are characterized by:
  • Randomness in their state transitions, making them unpredictable in the exact sequence of events but predictable in terms of long-term probabilities.
  • The use of mathematical models to describe processes that have inherent randomness, such as queueing systems, stock market movements, and natural phenomena.
Understanding stochastic processes is fundamental for predicting future states of systems in various fields, such as finance, physics, and artificial intelligence systems, all of which make extensive use of concepts derived from Markov chains.

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Most popular questions from this chapter

Consider a Markov chain with states \(0,1,2,3,4\). Suppose \(P_{0,4}=1\); and suppose that when the chain is in state \(i, i>0\), the next state is equally likely to be any of the states \(0,1, \ldots, i-1\). Find the limiting probabilities of this Markov chain.

(a) Show that the limiting probabilities of the reversed Markov chain are the same as for the forward chain by showing that they satisfy the equations $$ \pi_{j}=\sum_{i} \pi_{i} Q_{i j} $$ (b) Give an intuitive explanation for the result of part (a).

Let \(P^{(1)}\) and \(P^{(2)}\) denote transition probability matrices for ergodic Markov chains having the same state space. Let \(\pi^{1}\) and \(\pi^{2}\) denote the stationary (limiting) probability vectors for the two chains. Consider a process defined as follows: (i) \(X_{0}=1 .\) A coin is then flipped and if it comes up heads, then the remaining states \(X_{1}, \ldots\) are obtained from the transition probability matrix \(P^{(1)}\) and if tails from the matrix \(P^{(2)} .\) Is \(\left\\{X_{n}, n \geqslant 0\right\\}\) a Markov chain? If \(p=P\\{\) coin comes up heads \(\\}\), what is \(\lim _{n \rightarrow \infty} P\left(X_{n}=i\right) ?\) (ii) \(X_{0}=1 .\) At each stage the coin is flipped and if it comes up heads, then the next state is chosen according to \(P^{(1)}\) and if tails comes up, then it is chosen according to \(P^{(2)} .\) In this case do the successive states constitute a Markov chain? If so, determine the transition probabilities. Show by a counterexample that the limiting probabilities are not the same as in part (i).

A Markov chain is said to be a tree process if (i) \(P_{i j}>0\) whenever \(P_{j i}>0\), (ii) for every pair of states \(i\) and \(j, i \neq j\), there is a unique sequence of distinct states \(i=i_{0}, i_{1}, \ldots, i_{n-1}, i_{n}=j\) such that $$ P_{i_{k}, i_{k+1}}>0, \quad k=0,1, \ldots, n-1 $$ In other words, a Markov chain is a tree process if for every pair of distinct states \(i\) and \(j\) there is a unique way for the process to go from \(i\) to \(j\) without reentering a state (and this path is the reverse of the unique path from \(j\) to \(i\) ). Argue that an ergodic tree process is time reversible.

Let \(\left\\{X_{n}, n \geqslant 0\right\\}\) denote an ergodic Markov chain with limiting probabilities \(\pi_{i} .\) Define the process \(\left\\{Y_{n}, n \geqslant 1\right\\}\) by \(Y_{n}=\left(X_{n-1}, X_{n}\right) .\) That is, \(Y_{n}\) keeps track of the last two states of the original chain. Is \(\left\\{Y_{n}, n \geqslant 1\right\\}\) a Markov chain? If so, determine its transition probabilities and find $$ \lim _{n \rightarrow \infty} P\left\\{Y_{n}=(i, j)\right\\} $$
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