Transition Probability Matrix

[welch4369]

The 6 faces of a fair die are marked with the numbers 1,2,3,4,5,6. We roll the die repeatedly and independently, noting the numbers that come up. We say that Xn = j if j is the largest number that has come up in the first n tosses. (Define X0=1). The resulting DTMC has state space S={1,2,3,4,5,6}. Give its transition probability matrix.

So I understand the basics of how to set up a transition probability matrix, but for some reason this one is confusing me. Any help on setting it up would be great! Thank you!

[BGM]

The transition matrix is

\begin{bmatrix}
\frac {1} {6} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6}
& \frac {1} {6} \\[0.3em]
0 & \frac {1} {3} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6}
& \frac {1} {6} \\[0.3em]
0 & 0 & \frac {1} {2} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & \frac {2} {3} & \frac {1} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & 0 & \frac {5} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & 0 & 0 & 1\\[0.3em]
\end{bmatrix}

Sorry cannot type it out [LaTeX Error: String is too long (469, limit 400)]


Anyway the point is that the maximum process
is monotonic increasing, and it increase only when the new roll is greater,
and it will jump to that new maximum.
Otherwise, it will stay at the old value.

[welch4369]

That helps so much thank you!

[Martingale]

The transition matrix is

\begin{bmatrix}
\frac {1} {6} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6}
& \frac {1} {6} \\[0.3em]
0 & \frac {1} {3} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6}
& \frac {1} {6} \\[0.3em]
0 & 0 & \frac {1} {2} & \frac {1} {6} & \frac {1} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & \frac {2} {3} & \frac {1} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & 0 & \frac {5} {6} & \frac {1} {6} \\[0.3em]
0 & 0 & 0 & 0 & 0 & 1\\[0.3em]
\end{bmatrix}

Sorry cannot type it out [LaTeX Error: String is too long (469, limit 400)]