Math 166 Lecture Notes M.2
Section M.2 Regular Markov Chains
One of the many applications of Markov chains is in making long-range predictions.
It is not possible to make long-range predictions with all transition matrices. However,
for a particular group of transition matrices it is always possible. If a transition matrix is
“regular”, it will always be possible. A transition matrix is regular if some power of the
matrix contains all positive entries (no zeroes!).
A question that will soon come up is this…. How far must you go (with powers of the
matrix) to be certain that a matrix is not regular? The answer is that if zeros occur in the
same positions in two successive powers of a matrix, then they will appear in those
positions for all higher powers of the matrix.
Example 1: Is the transition matrix given below “regular”?
T =
.30.55.10
.10.20.65
.60.25.25
Find: T
50
=
_______________
_______________
_______________
Example 2: Is the transition matrix given below “regular”?
T =
0.8.35
.60.45
.4.2.2
Find: T
2
=
_________
_________
_________
, T
30
=
_______________
_______________
_______________
Example 3: Is the transition matrix given below “regular”?
T =
000.5
0.40.80.5
0.60.20
Find: T
2
=
_________
_________
_________
, T
3
=
____________
____________
____________
,
T
50
=
____________
____________
____________
Example 4: Is the transition matrix given below “regular”?
T =
000.5
0.610
0.400.5
Find: T
2
=
_________
_________
_________
, T
3
=
____________
____________
____________
,
Compute and comment on T
50
.
