Abstract

Stochastic matrices are a convenient way to model discrete-time and finite state Markov chains. The Perron–Frobenius theorem tells us something about the existence and uniqueness of non-negative eigenvectors of a stochastic matrix. In this entry, we formalize stochastic matrices, link the formalization to the existing AFP-entry on Markov chains, and apply the Perron–Frobenius theorem to prove that stationary distributions always exist, and they are unique if the stochastic matrix is irreducible.

BibTeX

@article{Stochastic_Matrices-AFP,
  author  = {Ren\'e Thiemann},
  title   = {Stochastic Matrices and the Perron-Frobenius Theorem},
  journal = {Archive of Formal Proofs},
  month   = nov,
  year    = 2017,
  note    = {\url{http://isa-afp.org/entries/Stochastic_Matrices.html},
            Formal proof development},
  ISSN    = {2150-914x},
}