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Finding long term Markov behaviors

  1. Aug 7, 2011 #1
    1. The problem statement, all variables and given/known data

    Find the long-term behavior of the Markov system described by the following matrix with initial conditions below: (i.e. find T^infinity)

    Find x_infinity

    2. Relevant equations

    x_0 = [.3 .3 .4]^T


    T =

    [.5 .4 .1]
    [.4 .3 .3]
    [.1 .3 .6]



    3. The attempt at a solution

    So in my attempt to find x_infinity I know that x_n = ( T^n)(x_0)

    This means:

    [.5^n .4^n .1^n] * [.3 .3 .4]^T
    [.4^n .3^n .3^n]
    [.1^n .3^n .6^n]

    Which I solved to be
    [ .3*.5^n .3*.4^n .4*.1^n]
    [ .3*.4^n .3*.3^n .4*.3^n]
    [ .3*.1^n .3*.3^n .4*.6^n]

    But as n approaches infinity all the terms get smaller and all go to zero. Does this seem correct so far?

    That would mean:

    x_infinity = ( 0 0 0 )^T ??? I tried this on my webassign, but it is not correct. Any tips or help in pointing out my mistakes would be great, thanks!
     
    Last edited: Aug 7, 2011
  2. jcsd
  3. Aug 7, 2011 #2

    Ray Vickson

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    Your expression for T^n is wrong. T^n = nth power of T in the sense of matrix multiplication. For example, T^2 = T*T =[[.42,.35,.23],[.35,.35,.31],[.23,.31,.46]], etc. You need to perform vector-matrix multiplications, or else solve the "equilibrium equations" to find the long-run limiting probabilities.

    RGV
     
  4. Aug 7, 2011 #3
    Oh I see how I was doing the expression of T^n was wrong. What do you mean by vector-matrix multiplcation or "equilibrium equations"? Do you mean keep finding T^1,T^2,T^3, and see if it approaches a limit?
     
  5. Aug 7, 2011 #4

    Ray Vickson

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    The state-probability row vector after n steps is x_n = x_0 * T^n, which can be written as x_{n-1}*T (a vector times a matrix). If the x_n approach a limiting distribution y, we must have y = y*T, and sum y(j) = 1. This is a simple system of linear equations, called the equilibrium or balance equations. I cannot believe that this material is not all in your textbook, but if not, Google "Markov chain" to see various relevant articles.

    RGV
     
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