# 2-state HMM / ARMA process?

I have issues with this problem:

Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian white noise.

Let $$Y_t = b_{X_t} + \sigma_{X_t} W_t$$ How to prove that $\{Y_t, t\in \Bbb N\}$ is an ARMA(1,1) process?

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## migrated from math.stackexchange.comMay 8 '14 at 14:17

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I thought that cross-posting has to be avoided. – mookid May 7 '14 at 20:12
You're right, it should be avoided. We can try try get a moderator to migrate it if you want. – joebloggs May 7 '14 at 22:10
So you have a Markov chain $X_t$ and as you describe the transition matrix $M$ it is possible to go from state $1$ to $2$ and back. Is this really enough info to prove that $Y_t$ solves $Y_t = a Y_{t-1} + b W_{t-1} + W_t$, the ARMA(1,1) representation? – Richard May 12 '14 at 9:27
I think it is enough to prove that it is the stationnary solution of this equation, indeed. One way suggested seems to look at the ACF of $Y$. – mookid May 13 '14 at 21:20