Question

2. A Markov chain is said to be doubly stochastic if both the rows and columns of the transition matrix sum to 1. Assume that the state space is {0, 1,....m}, and that the Markov chain is doubly stochastic and irreducible. Determine the stationary distribution T. (Hint: there are two approaches. One is to solve T P and ( 1 in general for doubly stochastic matrices. The other is to first solve a few examples, then make an educated guess about T, and finally show that your guess is correct.)

Answer 1

Solution:

Stationary distribution is a unique solution of the following two equation,

71

P.1) TPij i-0

Ti1(2) i-0

The summation is on the total no of states in sample space which is m+1 going from 0 to m.

It is doubly stochastic, therefore

....... (3)

ΠL ΣΡ- Ε Pij 27 i-0

Using equation 1 and 3 we get,

yν α Pij 1--ΣP -ΣΤΡ 1-TTj i-0 i-0

α (Pij 1-TTj i-0

α (Pij 1-TTj i-0

(1 - i)Pij 1-TTj i-0

Dividing by m on the both the sides we get,

.....(4)

(1 — ті) 1 - тj -Ра т т i-0

If $\pi_j$ is unique solution of equation 1 and 2 , then from equation 4 it is clear that

is also the solution. But those two have unique solution. so using uniqueness property we can say that,

1-Tj m

1 Tj m 1 i m 1 m 1 1 m1 Tj 1 m1