Question

An m × m stochastic matrix P is said to be doubly stochastic if its column sums are all equal to 1. If P is irreducible, what can be said about (i) the stationary distribution π and (ii) the time-reverse transition matrix Pˆ?

Answer 1

Here, it is given that the stochastic matrix P is a doubly stochastic matrix therefore by definition of doubly stochastic matrix

(i.e. row sum is equal to 1) also

(i.e. column sum is equal to 1)

And P if irreducible with m state then

(i) The stationary distribution is uniform distribution given by  

Proof :

Let,  

Therefore,

  

  

Which statistifies the condition for stationary distribution. Therefore the stationary disribution is uniform distribution.

ii) Now, we know that P is doubly stochastic and it's stationary distribution is uniform distribution.

Therefore, P statistifies

which, means P is reversible markov chain.

Therefore time reverse transition matrix will have same form as P.