Markov chain, sum of N dice rolls

[simba_]

Question : Let X_n be the maximum score obtained after n throws of a fair dice

a) Prove that Xn is a markov chain and write down the transition matrix

Im having a problem starting the transition matrix

im assuming the states are meant to be the sum. then do you write out the transition matrix for the first 2 throws and have this matrix to the power of n-1?

 

[Stephen Tashi]

im assuming the states are meant to be the sum

I suppose if you are studying markov chains with an infinite number of states, you could try interpreting "maximum score" to mean some sort of sum. However, it seems to me that the problem intends the state of the process on the nth roll to be max \{ R_1,R_2,...R_n\} and not R_1 + R_2 + ... + R_n. So if make 3 rolls and they are {3,5,4} the state of the process is X_3 = 5

 

[simba_]

Thanks for your reply, that makes sense.

So the transition matrix is an upper triangular matrix to the power of n-1 with the diagonal entries 1/6, 2/6, 3/6, 4/6, 5/6, 6/6 respectively?

 

[Stephen Tashi]

So the transition matrix is an upper triangular matrix to the power of n-1

That is incorrect terminology. To compute things about the state at step n in the process, one may raise the transition matrix to a power, but the transition matrix itself, in simple examples, is not a function of n.

with the diagonal entries 1/6, 2/6, 3/6, 4/6, 5/6, 6/6 respectively?

Yes.

 

[simba_]

Thank you for your help