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A stochastic matrix is called doubly stochastic if its rows and columns sum to 1. Show that a Markov chain whose transition matrix is doubly stochastic has a stationary distribution, which is uniform on the statespace.

I'm trying to under stand the related problem but I don't understand how the uniqueness theorem comes in in the first row of the accepted answer, shouldm't it be $\pi_i$? What do those sums even mean? They don't have any upper bound but i suppose the upper bound should be the upper bound of the statespace?

Parseval
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1 Answers1

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They're using the uniqueness theorem so that, when they show that the uniform distribution is a valid stationary distribution for the doubly stochastic transition matrix, it must be the stationary distribution.

The uniqueness of the stationary distribution comes directly out of the irreducibility of $P$ + aperiodicity of $P$ + finite state space of the system, and the sums are just matrix multiplications over the state space (from $i=0$ to $M$).

aghostinthefigures
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