Column stochastic matrix/Positive row/Convergence/Fact

Convergence theorem for stochastic matrices

Let ${}M$ be a column stochastic matrix, fulfilling the property that there exists a row in which all entries are positive.

Then for every

distribution vector ${}v\in \mathbb {R} _{\geq 0}^{n}$ with ${}\sum _{i=1}^{n}v_{i}=1$ , the sequence ${}M^{n}v$ converges to the uniquely determined stationary distribution

of

{}M

.

Proof, Write another proof