Page 1
Math 141-copyright Joe Kahlig, 10B
Section 9.2: Regular Markov Chains
Definition: A transition matrix (stochastic matrix) is said to be regular if some power of T has all
positive entries. This means that the Markov chain represented by T is called a regular Markov chain.
Example: Are the transition matrices regular?
A) T =
"
0.6
.4

0

C) T =  0
1
0.2
0.8
0.2
0.3
0.5

0.3

B) T =  0
0.7
#


1

0 
0



D) T = 
1
0
0
0
0.2
0.4
0.2
0.2
0
0
1

0.6

0.4 
0
0.2
0.1
0.2
0.5
0.1
0
0.5
0.4





Definition: A Markov process that does not have a change in the next distribution state has reached
a steady state or equilibrium state. The steady state is denoted by XL .
A Markov process that has a regular transition matrix will have a steady state.
Example: From the car/bus problem from 9-1. Find the steady state distribution.

car

T = car  0.7
bus
0.3

bus

0.1 
0.9
Page 2
Math 141-copyright Joe Kahlig, 10B
Example: Find the steady state for the pizza problem from 9-1.
T =
A
B
C





A
0.5
0.3
0.2
B
0.4
0.2
0.4
C
0.25
0.5
0.25





Definition: The limiting matrix or the stable matrix is is defined as L = lim T n .
n→∞