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26 October, 21:51

Show that any second-order Markov process can be rewritten as a rst-order Markov pro - cess with an augmented set of state variables. Can this always be done parsimoniously, i. e., without increasing the number of parameters needed to specify the transition model?

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  1. 26 October, 22:14
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    Let Yt be a variable that will be able to take this form y1, y2, y3 ..., xk. Markov Chain first - order property states that:

    P (Ut=u|Ut-1, Ut-2, Ut-3 ...) = P (Ut=u|Ut-1)

    while Markov Chain second - order property allows us to write:

    P (Ut=u|Ut-1, Ut-2, Ut-3 ...) = P (Ut=u|Ut-1, Ut-2)

    We can be able to change the second-order Markov Chain into the first-order Markov Chain by regrouping the state-space as follow: Let At-1, t be a variable that carries 2 consecutive states of the Zt variable, that is to say : If Zt can carry value z1, z2, z3 then we define At-1, t such that At-1, t can take either z1z1, z1z2, z1z3, z2z1, z2z2, z2z3, z3z1, z3z2, z3z3. In this new state-space we are going to have:

    P (At-1, t=at-1, t|At-1, t-1, At-2, t-1, At-3, t-2 ...) = P (At-1, t=at-1, t|At-1, t-1)
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