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Autocorrelation Function of Markov Chain/Discrete Process

  1. Apr 10, 2012 #1
    1. The problem statement, all variables and given/known data
    I'm working with a transition matrix that is
    1-p, p
    q, 1-q

    And I need to find
    [tex]E(X_n X_{n+\tau})[/tex]
    with E = {-1,1}. The first row is probabilities given currently in state -1.

    And this needs to be a function of tau. I started by initiializing my M.C. with the steady-state probabilities, thinking that would ensure X_n was stationary (and so its autocorrelation would be a function of only tau).

    I then thought there would be two cases:
    [tex]\tau = 0[/tex]
    [tex]E(X_n^2)=\pi_1 + \pi_{-1} = 1[/tex]

    Then the other case
    [tex]\tau \ne 0[/tex]
    [tex]E(x_n x_{n+\tau}) = (1)(1)\pi_1^2 + (-1)^2 \pi_{-1}^2 - 2 \pi_1 \pi_{-1} = \left ( \frac{p-q}{p+q} \right)^2[/tex]

    Problem:
    I simulated the MC and computed the autocorrelation. Things I found:
    1.) For tau = 0, I am right.
    2.) My value for tau != 0 is the case when tau > about 6 or 7.
    3.) For tau = 1,2,3 or so the values are consistently different from my solution. I need to know what is going on.
    4.) If p = 1- q, P has all equal rows. In this situation, my solution works.

    Confusions:
    I don't understand how I could be wrong since I initialized the beginning state using the steady-state probabilities. So I thought the probability of going to any state in n steps from there was:
    [tex] \pi P^n[/tex]
    evaluated at the state you want That gives you pi_state no matter what.

    I can post my code if wanted.
     
    Last edited: Apr 11, 2012
  2. jcsd
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