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Let $X_{n}$ be a Markov chain on state space $S = \{ 1,2 \dots, 23 \}$ with transition probability given by

$p_{i,i+1}= p_{i,i-1} = \frac {1}{2} \ \ \forall \ 2\le i \le 22 , $

$ p_{1,2}= p_{1,23} = \frac {1}{2} $

$ p_{23,1}= p_{23,22} = \frac {1}{2} $

then we need to show that $P(X_n=i) = \frac {1}{23} $.

attempt :

( i thought of many results that i know but i could not figure it out )

i tried to solve equations

$\pi_1 = \frac {1}{2} \pi_2 + \frac {1}{2} \pi_{23} $

$\pi_2 = \frac {1}{2} \pi_1 + \frac {1}{2} \pi_3 $

..

..

$ \pi_{23} = \frac {1}{2} \pi_1 + \frac {1}{2} \pi_{22} $

but this seems confusing. Please suggest a proper method .

Davide Giraudo
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ANUJ NAIN
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1 Answers1

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Note that $p(X_n=i)=\frac{1}{23}$ suggests that the distribution is independent of $n$ a result which You already used implicitly when you wrote $\pi P=\pi$ where $\pi$ is a vector and $P$ is the $23\times23$ transition matrix. This suggests that $\pi_n$ converge towards stationary distribution as $n\rightarrow \infty$ and that $\pi$ is the steady-state probability. Consider how You would compute $\pi$ as a result of infinite number of transitions. In particular, consider that $\pi_n=\pi_0 P^n$ and that $\lim_{n\rightarrow \infty} \pi_0 P^n= \lim_{n\rightarrow \infty} P^n= \pi$. You can then use the last equality to compute $\pi$ and to show that $\pi_i=\frac{1}{23}$.

matus
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  • yes please provide some more hint . i still didn't understand – ANUJ NAIN Apr 19 '17 at 04:25
  • @ANUJNAIN added details – matus Apr 19 '17 at 13:43
  • howcome $\lim_{n\rightarrow \infty} \pi_0 P^n $became $ \lim_{n\rightarrow \infty} P^n $ why does $\pi_0$ vanish ? ...... – ANUJ NAIN Apr 19 '17 at 18:20
  • See the wikipedia section on [steady state analysis](http://en.wikipedia.org/wiki/Markov_chain#Steady-state_analysis_and_limiting_distributions) for more information. – matus Apr 19 '17 at 18:55
  • i read it but i didn't found anything that can solve this., i still dont get how is getting $\frac{1}{23}$ , can you tell me why is it getting $\frac{1}{23}$ ? – ANUJ NAIN Apr 20 '17 at 05:28
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    I added a final small clarification. Your remaining difficulties are related to linear algebra not statistics. In particular, see [here](http://math.stackexchange.com/questions/843994/limit-of-a-matrix-multiplication) – matus Apr 20 '17 at 11:13