0
$\begingroup$

In Pinsky and Karlin's Introduction to Stochastic Modeling, it is said without a proof that every regular stochastic matrix has a limiting distribution. More precisely,

enter image description here

Does anyone have a reference for a proof of this statement?

  • 1
    This is simply Perron-Frobenius theorem for primitive matrices. – 2017-02-25
  • 0
    @Artem: Thanks for your comment! I didn't know that. I will come back latter once I figure out enough details. – 2017-02-25
  • 0
    A full proof of this theorem, without reference to Perron-Frobenius theorem, is given in the First course in Stochastic Processes by Karlin and Taylor, in Chapter 4. – 2017-02-25
  • 0
    @Artem: Thanks for the reference! I will check out that book to see details. I would accept your comments as an answer if you could post it. – 2017-02-25

1 Answers 1

1

This theorem is an immediate consequence of Perron-Frobenius theorem for primitive matrices (as given, e.g., Horn's Matrix Analysis).

A full proof of this theorem, without reference to Perron-Frobenius theorem, is given in the First course in Stochastic Processes by Karlin and Taylor, in Chapter 3.

(As a side remark I note that the book you are using is a "light" version of Karlin and Taylor two volume classic. I would suggest to switch to the original one, which is a quite deep exposition of the stochastic processes without any measure theory)

  • 0
    Would you elaborate which theorem are you referring to in Karlin and Taylor? I'm not able to find it in Chapter 4 of the second edition, which is about continuous time Markov chains. – 2017-02-26
  • 0
    @Jack It is Theorem 1.3 in Chapter 3. – 2017-02-26