Question

Initial state and limit state

In the class I have shown you with the population example of Pomona and Walnut with initial population for both equal to 100k, and limit state or steady state equal to 120K for Pomona and 80K for Walnut. The transition probabilities are 0.2 (or 20%) from Pomona to Walnut and 0.3 or 30% from Walnut to Pomona

This corresponds to the case of p(0) = [0.5, 0.5] and limiting state = [0.6, 0.4]. (see section 12.3, pages 450 – 451 for the notations).

Q: Does there exist a Markov chain M that has exactly the opposite, i.e. p(0) = [0.6, 0.4] with the limiting state

= [0.5, 0.5]? If so, show such example M (a 2x2 matrix) and prove that the limiting state is [0.5, 0.5]. If not, explain why it does NOT exist.

Answer 1

	Pomona	Walnut
Initial state	100k	100k
limit state	120K	80K
transition probabilities	0.2 (or 20%)	0.3 (or 30%)

p(0) = [0.5, 0.5] and limiting state = [0.6, 0.4]

using state diagram,

P₀₀ = 0.8 (1-p) P₀₁ = 0.2 (p) P₁₀ = 0.3 (q) P₁₁ = 0.7 (1-q)

there are 2 ways this can happen

P₀₀ P₀₁ = 0.8 * 0.2 = 0.16

P₀₁ P₁₁ = 0.2 * 0.7 = 0.14

in two steps is,

P₀₀ P₀₁ + P₀₁ P₁₁ = 0.16 + 0.14 = 0.3

the state transition matrix for this Markov chain is

P = 0.5 0.5

0.6 0.4

if we square this matrix we get,

P² = 0.55 0.45

0.54 0.64

no such Markov chain exists, since, P₀₁ of the P² matrix is not equal to the 0.3 (P₀₀ P₀₁ + P₀₁ P₁₁ )  value obtained.