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is there a way to construct a transition matrix of a time-homogeneous, finite Markov chain such that the stationary distribution always has full support (this is equivalent to all states of the chain being positive recurrent).

I am looking for something quick that doesn't involve drawing the state-transition graph and checking for absorbing states etc.

Best would be something on the level of "check that every column sums to one". Is this just wishful thinking?

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When you write "the stationary distribution, I suppose you mean that the Markov chain is irreducible. An easy sufficient condition is that all entries are strictly positive (and that the sum of each row is $1$ so it is a transition matrix).

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    $\begingroup$ For a sparser pattern, tridiagonal with strictly positive entries on the sub- and superdiagonal is also fine. The main result to use is the Perron-Frobenius theorem. $\endgroup$
    – Federico Poloni
    Jul 28, 2014 at 6:29

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