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What are the biggest transitions after matrix

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In a transition matrix, is the biggest component of the stationary distribution the one that correspond to the column with the biggest entries-sum?

markov-chains

Given a transition matrix, is the biggest component of the stationary distribution the one that correspond to the column whose sum of entries is the biggest among all columns?

(By "correspond" I mean The ith component of the vector↔the ith column of the matrix)

It seems plausible since in the transition matrix the column with the biggest sum of the entries is the most "absorbing" state. My probability teacher told me he often asks his colleagues about this fact, never hearing it is a known fact.

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Oct 3 '11 at 20:30

Immanuel Weihnachten

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Here's another counterexample that may throw some light on why this isn't the case:

P=⎛⎝⎜⎜⎜1−2ϵ001ϵ000ϵ0000110⎞⎠⎟⎟⎟,π≈(1−4ϵ,ϵ,ϵ,2ϵ).

The last column has sum 2, the first one 2−2ϵ, but the stationary distribution is concentrated almost entirely in the first state. This shows that it's not just the probabilities of transitions to a state that determine its stationary probability, but also the probabilities of transitions away from that state, and the stationary probabilities of the states that lead to it. In the example, though the first and last state have very similar probabilities for transitions to them, the first state has a much lower probability for a transition away from it.