Imagine a two-state Markov chain which hops between the states $\pm 1$ with probability $p<1/2$, so that the autocorrelation function after $k$ steps is

$\rho_k = (2p-1)^k$

If I take an exponential moving average of this series with weighting parameter $\lambda$, what does the distribution of values of the new series look like?

Probably the answer is "gaussian, centered on 0" but what is the variance? Is there a known result that makes this computation trivial?