# Powers of Markov transition matrices

• cientifiquito
In summary, the theorem states that the ijth entry of the transition matrix Pn gives the probability that a Markov chain, starting in state si, will be in state sj after n steps. This can be proven by induction, using the equation p(2)ij = \sum^{r}_{k=1}pikpkj, where r is the number of states in the Markov chain and P is the square matrix with ik being the probability of transitioning from i to j. Using the Einstein index notation shortcut, we can simplify the process of finding the state vector at time n to s^{(n)} = s^{(0)}P^n. However, it is important to note that P may not always be a
cientifiquito

## Homework Statement

Prove the following theorem by induction:
Let P be the transition matrix of a Markov chain. The ijth entry p(n)ij of the matrix Pn gives the probability that the Markov chain, starting in state si, will be in state sj after n steps.

## Homework Equations

p(2)ij = $\sum^{r}_{k=1}$pikpkj
(where r is the number of states in the Markov chain and P is the square matrix with ik being the probability of transitioning from i to j)

## The Attempt at a Solution

assume that

p$^{(n)}_{ij}$ = $\sum^{r}_{k=1}$p$^{(n-1)}_{ik}$p$^{(n-1)}_{kj}$

then pn+1 must be:
p$^{(n+1)}_{ij}$ = $\sum^{r}_{k=1}$p$^{(n +1 - 1)}_{ik}$p$^{(n + 1 -1)}_{kj}$

that's all I've come up with but it doesn't convince me very much

i think your first step isn't quite right

so say P is your transition matrix and $$s^{(n)}$$ is your state vector at time n, the ith element is the probability of being in state i:
$$s^{(1)} = s^{(0)}P$$
$$s^{(2)} = s^{(1)}P= s^{(1)}P = s^{(0)}PP= s^{(0)}P^2$$
$$s^{(3)} = s^{(2)}P= s^{(1)}PP = s^{(0)}PPP= s^{(0)}P^3$$

then generalise to $$s^{(n)}$$

this should be pretty easy to wirte in index form

Last edited:
Also do you know the einstein index notation short cut?

repeated indicies in a product are summed over, saves a bit of time
$$p_{ij}p_{jk} = \sum_j p_{ij} p_{jk}$$

And one more, just be a bit careful with you index representation, generally P is not a symmetric matrix (may want to check this) so writing the product PP needs a bit of thought, though i think you've got it right

Last edited:

## 1. What is a Markov transition matrix?

A Markov transition matrix is a square matrix that represents the probabilities of transitioning from one state to another in a Markov chain. Each row of the matrix represents the current state, while each column represents the next possible state. The values in the matrix indicate the probability of transitioning from the current state to the next state.

## 2. How is a Markov transition matrix calculated?

A Markov transition matrix is calculated by dividing the number of times a transition occurs by the total number of transitions for each state. This results in a matrix of probabilities that sum to 1 for each row. The matrix can also be calculated by using a mathematical formula known as the Chapman-Kolmogorov equation.

## 3. What is the significance of a Markov transition matrix?

Markov transition matrices are significant because they allow for the prediction of future states in a Markov chain. This is useful in various fields such as economics, genetics, and weather forecasting. By analyzing the probabilities in the matrix, one can determine the long-term behavior of a system or make predictions about future events.

## 4. Can a Markov transition matrix have negative values?

No, a Markov transition matrix cannot have negative values. Since the values in the matrix represent probabilities, they must be between 0 and 1. Negative values would not make sense in this context and would result in invalid calculations.

## 5. How can the accuracy of a Markov transition matrix be improved?

The accuracy of a Markov transition matrix can be improved by increasing the number of transitions or observations used to calculate it. This can be done by collecting more data or by using a larger time frame. Additionally, the matrix can be recalculated periodically to account for any changes in the system being modeled.

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