Let $A$ be a primitive matrix and let $\lambda$ be its Perron-Frobenius eigenvalue. We give formulas expressing the associated normalized Perron-Frobenius eigenvector as a simple functional of a multitype Galton-Watson process whose mean matrix is $A$, as well as of a multitype branching process with mean matrix $e^{ (A-I)t}$. These formulas are generalizations of the classical formula for the invariant probability measure of a Markov chain.
Categories: math.PR