What do you think the probability vector is after 20 transitions of P?

Question

Consider the following Markov chain

0 1/2 1/2

P=3/10 7/10 0

1/2 0 1/2

Probability Vector:

W [ 3/11, 5/11, 3/11]

What do you think the probability vector is after 20 transitions of  P?

Answer 1

This will be approximated by the eigenvector for the eigenvalue with eigenvector = 1

Let z = 1

From the second column, 1/2x + 7/10y = y

From the third column, 1/2x + 1/2 = 1

Rewriting,

1/2x - 3/10y = 0

1/2x = 1/2

x = 1

(1/2)(1) - 3/10y = 0

1/2 = 3/10y

y = 5/3

(1, 5/3, 1) normalizes to

(3/11,5/11,3/11)

With the initial probability vector equal to the eigenvector, there will be the same values at every iteration.