We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B$ all of whose rows are equal to $P ^ { \infty } = \left( b _ { 1 } , \ldots , b _ { n } \right)$. Prove that the sequence $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } } = \left( P ^ { ( 0 ) } M ^ { k } \right) _ { k \in \mathbb { N } }$ converges to $P ^ { \infty }$, regardless of the initial probability distribution $P ^ { ( 0 ) }$.
We now assume that all coefficients $m _ { i , j } ( 1 \leqslant i , j \leqslant n )$ of the stochastic matrix $M$ are strictly positive. The sequence $\left( M ^ { k } \right)$ converges to a stochastic matrix $B$ all of whose rows are equal to $P ^ { \infty } = \left( b _ { 1 } , \ldots , b _ { n } \right)$.
Prove that the sequence $\left( P ^ { ( k ) } \right) _ { k \in \mathbb { N } } = \left( P ^ { ( 0 ) } M ^ { k } \right) _ { k \in \mathbb { N } }$ converges to $P ^ { \infty }$, regardless of the initial probability distribution $P ^ { ( 0 ) }$.