Probabilistic approach to Perron root, the group inverse, and applications

Abstract

A probabilistic approach to the study of the Perron root of irreducible nonnegative matrices is presented. Our two main results are a probabilistic representation for the generalized inverse of the generator of a continuous-time finite-state Markov chain and the identification of the Hessian of the Perron root of a nonnegative irreducible matrix, with respect to a certain natural transformation of its entries, as a covariance matrix for additive functionals for a related Markov chain. These provide us with a natural approach – at least from the point of view of the probabilist – to study positivity and convexity properties of perturbations to the Perron eigenvalue. We focus on reestablishing and improving a number of known results in the field.

Keywords:

• additive functional
• central limit theorem
• group inverses
• Markov chains
• nonnegative matrices
• Perron roots

AMS Subject Classifications::

• 15A48
• 15A09
• 60G44
• 60J22
• 65C40