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Limiting Distributions

+1 vote
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asked Apr 7, 2017 in STA 3041F - Time Series and Markov Chains by ChanceTheRapper (650 points)

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??

1 Answer

+2 votes
answered Apr 8, 2017 by asilmotala (2,610 points)
selected Apr 12, 2017 by Rowan
 
Best answer

Firstly, thanks for including the actual question in your post it is highly useful.


Secondly, this question is actually taken from an overseas university. Please see the attached link for a detailed explanation of the question and solution.

http://www.uio.no/studier/emner/matnat/math/STK2130/v15/ukeoppgaver/oppg20og22.pdf

I hope this helps!

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