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Math Help - Limiting distribution = Initial Distribution MC

  1. #1
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    Limiting distribution = Initial Distribution MC

    In a Markov chain, if the limiting distribution is identical to the initial Distribution, then what does this say about the distribution of Xn? (the n is a subscript).

    Thanks!
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  2. #2
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    Quote Originally Posted by icobes View Post
    In a Markov chain, if the limiting distribution is identical to the initial Distribution, then what does this say about the distribution of Xn? (the n is a subscript).

    Thanks!
    It is constant.

    CB
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    Thanks! Could you please explain why this is the case.
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  4. #4
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    Quote Originally Posted by icobes View Post
    Thanks! Could you please explain why this is the case.
    Because the limiting distribution is by necessity a stationary state. After a large number of steps the siatribution is A^n x \approx y, where $$ A is the transition matrix, $$n the number of steps, $$x the initial distribution and $$y the limiting distribution.

    So:

    Ay \approx A(A^n x) = A^{n+1} x \approx y

    (In fact because of the limits involved the left most term is identical to the right most term and not just approximatly equal, this could be made explicit by putting the appropriate limits in the above)


    CB
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