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Markov chain help

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    Just a quick couple of questions:
    What is meant by the 'equilibrium distribution vector' concerning a Markov chain of probabilities?

    Take this example:


    "An object moves along the graph shown above, so that at each step it moves with equal probability to a neighbouring point."

    a) Write down the transition matrix P for this Markov process (i think i've got this right, so i'm not bothered about this one)
    b) Calculate the equilibrium distribution vector for this Markov process.

    How do you calculate the limit as n -> infinity of A^n for a square matrix A?
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    From my rather hazy memory, an equilibrium (or invariant) distribution vector is a stochastic row vector \pi with \pi P = \pi.
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    (Original post by generalebriety)
    From my rather hazy memory, an equilibrium (or invariant) distribution vector is a stochastic row vector \pi with \pi P = \pi.
    Thanks, that makes sense now.
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    In general, finding \lim_{n\to\infty} A^n is tricky (even assuming the limit exists).

    But barring a few pathological cases(*), if M is a stochastic matrix, and \pi is an invariant vector for M, then M^n x \to \pi \for all x. Which means M^n converges to the matrix with all columns equal to \pi.

    (*) I think it always works as longs as M has no elements of size 1, but that's intuition and hope speaking rather than detailed knowledge.


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