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Markov chain help

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    Just a quick couple of questions:
    1
    What is meant by the 'equilibrium distribution vector' concerning a Markov chain of probabilities?

    Take this example:

    1---------2----------3

    "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.

    2
    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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Updated: February 12, 2009
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