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Limit for Markov Chains

  1. Mar 11, 2012 #1
    1. The problem statement, all variables and given/known data


    [itex]P=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)[/itex]

    Show that [itex]P^n[/itex] has no limit, but that: [itex]A_n=\frac{1}{n+1}(I+P+P^2+\ldots+P^n)[/itex] has a limit.

    3. The attempt at a solution

    I can see that [itex]P^{EVEN}=\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)[/itex] and [itex]P^{ODD}=\left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)[/itex], so a steady state is never reached, but I can't figure out the second part.

    Any suggestions?
  2. jcsd
  3. Mar 11, 2012 #2


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    Staff Emeritus
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    Gold Member

    So for example

    I+P+P2+P3+P4+P5 =
    [3 3]
    [3 3]

    and when you divide this by six you get a matrix with all 1/2s. Try adding up some more guys and see what happens
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