# Limiting Distributions

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Question 2b of 2016 test 1:

2. (a) A transition probability matrix P is said to be doubly stochastic if the sum over each column equals one, i.e, P i pij = 1, for all states j. If such a Markov Chain is irreducible and aperiodic and consists of M + 1 states 0, 1, . . . , M, show that the limiting probabilities are given by πj = 1 M + 1 , for j = 0, 1, . . . , M. [3]

(b) Let Yn be the sum of n independent rolls of a fair die. Find lim n→∞ P(Yn is a multiple of 13) .

Hint: Define an appropriate Markov Chain and apply part (a).

In (b) I don't understand how to start this. Is the die rolled several times until you get to 13? And then you basically check each time if it is a multiple of 13 and then subtract it?

Like if I roll a 6 and a 5 and a 4 then do I say Yn = 1 and then start rolling again where I now have (6+5+3)-13 = 2 left over to add the next rolls of the die to?

Also - if I am understanding the Q correctly, I still don't see how it's a MC??