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Say I have a transition matrix $Q = \begin{bmatrix} 1-p & p \\ p & 1-p \end{bmatrix}$ where $0 < p < 1$ for a two state system with states $-1$ and $1.$ Define $X_i$ to be the value of the markov chain at time $i$ (so either $-1$ or $1$). If $\bar{X_i} = \frac{1}{n}\sum_{i = 1}^n X_i$ what is the $\mathbb{E}[\bar{X_i}]$ and $Var[\bar{X_i}]$?

I've started off by tackling $\mathbb{E}[\bar{X_i}]$ but it seems to me that this answer depends on whether $n$ is even or odd. Note since this is a Markov chain there is not pairwise independence.

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At this question the stationary probabilities of such a Markov chain are calculated for the state transition matrix

$$\begin{bmatrix} p&1-p\\ 1-q&q \end{bmatrix}.$$ If $p=q>0$ then the $n^{th}$ power of the state transition probability tends quite fast to

$$\begin{bmatrix} \frac12&\frac12\\ \frac12&\frac12 \end{bmatrix}.$$

So, for $p=q$ the stationary probabilities are both $\frac12$.

So, on the long run $1$ and $-1$ appears with the same probability. The expectation is then $0$. As far as the variance, we have $1^2\frac12+(-1)^2\frac12$=1.

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