How to compute stationary distribution for martrix with more than 1 closed class

[sam48]

Hi,

Thank you in advance if anyone can answer this question.

How any stationary distributions exists in below matrix and what are the value

[
.5 0 0 .5
.25 .5 .25 0
0 0 1 0
1/6 0 0 5/6
]

Any information regarding how to compute stationary distribution in a martrix with more than 1 closed class would be appreciated.
regards,

 

[statdad]

Your chain has an absorbing state (the third state, since the 3,3 entry is 1). Write this matrix in canonical form and analyze it with the standard methods.

 

[bpet]

One way is just to solve the eigenvector equation xA=x - since your example has 2 closed classes (the other being {1,4}), there will be two LI eigenvectors. Positivity and summation to 1 will tell you which linear combinations are valid.