I have been told that a Markov process $X_1,X_2,\dotsc$, where $X_i\in \{1,-1\}$ with correlation $\rho$ satisfies $$ \mathbb{E}(X_i\mid X_{i-1})=\rho X_{i-1} $$ But am I correct in saying that this does not necessarily imply $\mathrm{corr}(X_{i},X_{i-1})=\rho$?
My reasoning is as follows: I get $\mathbb{E}(X_iX_{i+1})=\rho$ and $\mathbb{E}(X_i)=\rho\mathbb{E}(X_{i-1})$, so that $$\mathbb{E}\left[(X_i-\mathbb{E}(X_i))(X_{i-1}-\mathbb{E}(X_i))\right]=\rho-\rho\mathbb{E}(X_{i-1})^2$$ and $$\mathrm{Var}(X_i)=1-\rho^2\mathbb{E}(X_{i-1})^2,\qquad \mathrm{Var}(X_{i-1})=1-\mathbb{E}(X_{i-1})^2$$ and so $$\mathrm{corr}(X_{i},X_{i-1}) = \frac{\rho(1-\mathbb{E}(X_{i-1})^2)}{\sqrt{(1-\rho^2\mathbb{E}(X_{i-1})^2)(1-\mathbb{E}(X_{i-1})^2)}}=\rho\cdot \sqrt{\frac{1-\mathbb{E}(X_{i-1})^2}{1-\rho^2\mathbb{E}(X_{i-1})^2}}\neq \rho$$