Reference no: EM133155151

Question: Consider the following one-step transition probability matrices of two Markov chains:

 

(a) Use R to calculate P₁ⁿ with consecutive large n , such as n=1000, 1001, 1002,....
Find the following properties of the Markov chain with one-step transition probability matrix P₁ based on the results of R together with the theory of Markov chains:
• The periodicity of the states;
• The stationary distribution(s), if not unique, show at least two examples;
• The long-term properties: convergence of P₁ⁿ as n→∞;
• The conditions of the initial distributions Π⁽⁰⁾ such that Π⁽⁰⁾P₁ⁿ converges as n→∞ even if P₁ⁿ divergs;
• An example of Π⁽⁰⁾ with positive elements only (no 0) such that Π⁽⁰⁾P₁ⁿ converges as n→∞ and the limit of Π⁽⁰⁾P₁ⁿ

(b) Repeat part (a) for transition probability matrix P₂.

Notes:
• This question is focused on using R to draw sensible conclusions about Markov chains.
• Explain or justify your answers by the results of R and/or the theory of Markov chains.
• Formal mathematical proofs are NOT required.
• Provide relevant R-codes and outputs that lead to your answers.