Let $T$ be an $n \times n$ transition matrix, i.e. the rows sum to 1 and the entries all lie in the interval $[0,1]$. What are necessary and sufficient conditions for which the limit $lim_{n \rightarrow \infty} T^{n}$ exists?
I recall reading somewhere (but don't remember where) that this is true if and only if $1$ is an eigenvalue of $T$ with geometric multiplicity 1, and all other eigenvalues have length strictly less than 1. But this does not seem to be true, for the following matrix:
[0, 0.5, 0.5 , 0 , 0 , 0 , 0 ],
[0, 0 , 0.33, 0.33, 0.33, 0 , 0 ],
[0, 0 , 0 , 1 , 0 , 0 , 0 ],
[0, 0 , 0 , 0 , 0.5 , 0.5, 0 ],
[0, 0 , 0 , 0 , 0 , 0 , 1 ],
[0, 0 , 0.5 , 0 , 0 , 0 , 0.5],
[0, 0 , 0 , 1 , 0 , 0 , 0 ]
According to MATLAB, the limit of its powers appears to converge to:
0 0 0.1663 0.1650 0.0825 0.0825 0.4988
0 0 0.0825 0.3300 0.1650 0.1650 0.2475
0 0 0 1.0000 0 0 0
0 0 0 0 0.5000 0.5000 0
0 0 0.2500 0 0 0 0.7500
0 0 0.2500 0 0 0 0.7500
0 0 0 1.0000 0 0 0
Yet its eigenvalues are $1,0,-0.5000 + 0.8660i,-0.5000 - 0.8660i$ the last two of which have length strictly equal to 1.