Are any general conditions known on a finite transition nxn matrix that ensure that there exists at least one mth root which is also a transition matrix? It is easy to construct a 3x3 , diagonally dominant, transition matrix with distinct, real eigenvalues but no transition matrix, square root.
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No, as far as I know it is an open problem; there are no simple general conditions. You can find some work on the problem, including sufficient criteria, counterexamples and some plots that give heuristics, on http://www.maths.manchester.ac.uk/~lijing/papers/hili11.pdf 

