# Markov chains

1. Apr 21, 2009

### bigplanet401

1. The problem statement, all variables and given/known data

Let $$X$$ and Y be Markov chains on the set Z of integers. Is the sequence Z_n = X_n + Y_n necessarily a Markov chain?

2. Relevant equations
Markov property:

P[X_n=x_n | X_{n-1}=x_{n-1},...,X_0=X_0] = P[X_n=x_n|X_{n-1}=x_{n-1}]

3. The attempt at a solution

I'm thinking it is, because P[X+Y|Z] = P[X|Z]+P[Y|Z], and both X and Y have the Markov property; moreover, P[X|Z] = P[X|X_{n-1}, ..., X_0] because X doesn't care about Y in the first place. Does this seem right?

Last edited: Apr 22, 2009