Question

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

Short answer

Given that the transition probability matrix P of a Markov chain has the property that for some positive integer r, \( P^r \) has all positive entries. We know that n ≥ r, so we can write n as r + (n - r). By using matrix multiplication properties, we have: \( P^{n} = P^{r+(n-r)} = P^{r}P^{n-r} \). Since P is a transition probability matrix, it has all non-negative entries. As n increases, the probability of transitioning to a specific state also increases, so \( P^{n-r} \) will have all positive entries when n ≥ r. Using the theorem on the product of positive matrices, since \( P^r \) and \( P^{n-r} \) both have all positive entries, their product \( P^{n} \) will also have all positive entries for all integers n ≥ r.

Step-by-step solution

Analyzing given information

We are given that the transition probability matrix P of a Markov chain has the property that P^r has all positive entries for some positive integer r. This tells us that all elements in P^r will have a value greater than 0.

Multiplying P^r with P^n-r

In order to find the matrix P^n, we need to think about how we could get P^n in terms of P^r. Since we are given that n ≥ r, we can write n as r + (n - r). So, we can rewrite P^n as P^(r+(n-r)). By using matrix multiplication properties, we can say: \( P^{n} = P^{r+(n-r)} = P^{r}P^{n-r} \)

Analyzing the product of positive matrices

As we know that P^r has all positive entries, we will use the theorem for the product of positive matrices. The theorem states that if A and B are two matrices with all positive entries, then the product AB will also have all positive entries.

Analyzing the properties of P, a transition probability matrix

Since P is the transition probability matrix of a Markov chain, its rows sum to one. This means that there is a positive probability for each state to transition to any other state. Consequently, as n increases, the probability of transitioning to a specific state from another state also increases.

Using the theorem on P^r and P^n-r

By the theorem established in Step 3, we know that if P^r and P^n-r both have all positive entries, then the product P^r * P^n-r (which is P^n) will also have all positive entries. Since P is a transition probability matrix, we already know that P has all non-negative entries. Since multiplying P^n-r by P will only increase the probability of reaching a specific state (as mentioned in Step 4), P^n-r will also have all positive entries when n ≥ r.

Concluding the proof

Given that P^r has all positive entries and P^n-r has all positive entries when n ≥ r, by the theorem on the product of positive matrices mentioned in Step 3, we can conclude that P^n will have all positive entries for all integers n ≥ r.

Key concepts

Transition Probability Matrix

The backbone of any Markov chain can be visualized through its transition probability matrix. Simply put, this matrix, often denoted as \( P \), holds the probabilities that one state will transition to another state. For instance, if our Markov chain represents the weather, the matrix could contain the probability of it being sunny tomorrow given that today is rainy.

It's critical to understand that each row in \( P \) must sum up to 1, as the probabilities of transitioning from any given state to all possible states must collectively be certain — that is, one of those transitions must happen. This constraint on the rows being a probability distribution is what makes \( P \) a special kind of matrix in the realm of stochastic processes.

Markov Chain Properties

Markov chains have distinct properties that set them apart from other stochastic processes. Firstly, they have the memoryless property, meaning the future state depends only on the current state and not on the sequence of events that preceded it. This is known as the Markov property.

Furthermore, regularity is an important concept in Markov chains. A Markov chain is said to be regular if there exists some positive integer \( r \) such that all entries in the matrix \( P^r \) are positive. This indicates that it's possible to go from any state to any other state in \( r \) steps, making the states 'communicate' with each other effectively, which typically leads to long-term stability in the transition probabilities.

Matrix Multiplication

The power of matrix multiplication does wonders in understanding the dynamics of Markov chains. When you multiply a matrix by itself, what you’re doing is essentially chaining probabilities—finding out the likelihood of transitioning from one state to another over two time steps instead of just one. Mathematically, if you have \( P^r \), the matrix that tells you transition probabilities after \( r \) steps, then to find out the probabilities after \( n \) steps, where \( n > r \), you’d multiply \( P^r \) by \( P^{n-r} \). Doing so adheres to our principles of matrix multiplication where the row of one matrix fuses with the column of another, delivering a single entry in the product matrix that conveys a combined transition probability.

Positive Matrices Theorem

The Positive Matrices Theorem is crucial when dealing with transition probabilities in Markov chains. This theorem tells us that if you multiply two matrices that are strictly positive, meaning all their entries are more than zero, the resulting matrix will also be strictly positive.

In the context of Markov chains, this finding is significant because knowing that \( P^r \) has all positive elements assures us that any further powers of \( P \), that is any \( P^n \) with \( n \) being larger than \( r \), will also have all positive entries. It implies that as the system evolves, transitions between states are not just possible, they are guaranteed, solidifying the intrinsic interconnectivity among states within the chain.