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Homework Help: Coupling Markov chains

  1. Apr 28, 2009 #1
    1. The problem statement, all variables and given/known data
    Now that LaTeX is working again, maybe I can make the problem a little clearer.

    Let X and Y be Markov chains on the set [tex]\mathbb{Z}[/tex] of integers. Is the sequence [tex]Z_n = X_n + Y_n[/tex] necessarily a Markov chain?

    2. Relevant equations
    Markov property; coupling of Markov chains.

    3. The attempt at a solution

    I'm thinking no, except in the case where X and Y are independent. Otherwise

    \mathbb{P}[X_{n+1}=x_{n+1} | \mathcal{F}_Z] \neq \mathbb{P}[X_{n+1}=x_{n+1} | \mathcal{F}_X] \, ,

    where [tex]\mathcal{F}[/tex] is a filtration wrt the variable indicated.
  2. jcsd
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