7.45. Let Xn be an irreducible recurrent Markov chain of period d≥ 2. Let P be its transition probability matrix and Pd its d-step transition matrix. Show that the state space is a union of d disjoint sets Eo,... Ed-1 such that (a) Each E is an equivalence class for Pd. (b) If Xo Є Eo, then X₁ = E₁, X2 Є E3,...,Xd-1 € Ed-1 and Xd Є Eo.
7.45. Let Xn be an irreducible recurrent Markov chain of period d≥ 2. Let P be its transition probability matrix and Pd its d-step transition matrix. Show that the state space is a union of d disjoint sets Eo,... Ed-1 such that (a) Each E is an equivalence class for Pd. (b) If Xo Є Eo, then X₁ = E₁, X2 Є E3,...,Xd-1 € Ed-1 and Xd Є Eo.
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 18EQ
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