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Suppose we have $N$ independent 2-point Markov chains each having a rate matrix $Q = [-1,1;1,-1]$ and stationary distribution $\pi = [0.5,0.5]$. At time $t=0$, we initiate the chains so that the empirical measure $\approx \pi$. If $N$ is big, as time progresses the empirical measure will remain somewhere around $\pi$, but will exhibit some fluctuations.
For a 2-point chain, the empirical measure can be described by a single parameter, say $x$ which fixes the distribution at $[(1-x)/2, (1+x)/2]$. The initial condition says that $x(0) \approx 0$.

My question is - Is there any 'scaling law' that describes the fluctuations of $x$, as $N$ tends to $\infty$? Here by 'describes' I mean that the scaled process should converges in distribution to, say a brownian motion or solution to some appropriate stochastic differential equation with initial state $0$.

I apologise for the vague nature of the question, I'm not an expert in this area. Any pointers to references will by appreciated!

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The model of $N$ independent 2-point Markov chains as you have describes is essentially what is known as the Ehrenfest urn chain. Much is known about this Markov chain and you can probably find what you're interested in already in the literature.

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