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,
Does anyone have a reference for a proof of this statement?
2026-05-16 18:24:03.1778955843
Reference Request: Why do regular stochastic matrices have limiting distributions?
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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)