Appendix A: power of a sub-stochastic matrix
Proposition 14. If P is an order L irreducible sub-stochastic matrix admitting a row-eigenvector basis (e1,...,eL) where each ej, is associated to the eigenvalue λj and |λ1| ≥ |λ2| ≥...≥ |λL| then we have
where λ = |λ1| = λ1, P∞ =  and ∀j
Proof. For any vector  we have
As P is an irreducible, Perron-Frobénius theorem (see [3] for example) assure that λ is real and the sub-stochastic property, that λ ≤ 1 (in the particular case where P is also primitive i.e. ∃m, Pm > 0 then |λ2| <λ) Replacing x by Iℓ equation (51) gives the expression of the row ℓ of Pi and the proposition is then proved.     □
Corollary 15. If P is an order ≥ 2 irreducible sub-stochastic matrix admitting a diagonal form then there exists a matrix P∞ such as
Pi - λiP∞ = O(νi)     (52)
uniformly in i and where 1 ≥ λ ≥ ν are the two largest eigenvalues magnitudes. In the special case where P is primitive, then λ > ν and P∞ = limi→+∞ Pi/λi is a positive matrix.
Proof. Using proposition 14 we get
and the corollary is proved.     □