At the end of 2 nd week the state vector is
Px 1

x (2) =
Px (1)
=
|.25
.20
.25
.30|
|.25|
=
|.2550 |
|.20
.30
.25
.30|
|.20|
=
|.2625 |
|.25
.20
.40
.10|
|.25|
=
|.2325 |
|.30
.30
.10
.30|
|.30|
=
|.2500 |

Note that we can compute x 2 directly using
x 0 as
x (2) = Px (1) = P(Px (0) ) =
P 2 x (0)
Similarly, we can find the state vector for 5 th , 10 th , 20 th , 30 th , and 50 th observation periods.
x (5) =
P 5 x (0) =
.2495
.2634
.2339
.2532
x (10) =
P 10 x (0) =
.2495
.2634
.2339
.2532
x (20) =
P 20 x (0) =
.2495
.2634
.2339
.2532
x (30) =
.2495
.2634
.2339
.2532

x (50) =
.2495
.2634
.2339
.2532

The same limiting results can be obtained by solving the linear system of equations P P = P using this JavaScript. It suggests that the state vector approached some fixed vector, as the number of observation periods increase. This is not the case for every Markov Chain. For example, if
P =
0
1
1
0
, and