# Coupling Markov chains

1. Apr 28, 2009

### bigplanet401

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 $$\mathbb{Z}$$ of integers. Is the sequence $$Z_n = X_n + Y_n$$ 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 $$\mathcal{F}$$ is a filtration wrt the variable indicated.