Week 5 - Markov processes, Second order recurrence equations

Question 1

2 criteria for an nxn matrix A to be the TRANSITION MATRIX of a Markov process

Answer 1

1. The entries of A are all NON-NEGATIVE
2. The SUM of the entries in each column of A is 1

Question 2

Theorem for Markov process

Answer 2

1. If A is the transition matrix of a Markov process, then λ=1 is always an EIGENVALUE of A & all the other eigenvalues of A will satisfy |λ| <=1
2. If all entries of A are POSITIVE, there will only be 1 eigenvector for λ=1 & all other eigenvalues of A will satisfy |λ| <1

Question 3

What is the implication if all entries of transition matrix A are POSITIVE?

Answer 3

The sole λ=1 eigenvector will determine the LONG-TERM distribution of the population

Question 4

3 cases for solving 2nd order recurrence equations (SOREs)

Answer 4

1. 2 distinct real solutions
2. 1 repeated real solution
3. No real solution

See notes for yt equation forms

Question 5

3 steps for solving non-homogeneous SOREs

Answer 5

1. Solve HOMOgeneous equation to get Complementary sequence
2. Find the Particular sequence which satisfies non-homog. SORE
3. Combine Xt = CS + PS