Maybe this can be done by hand, but if not, it is a consequence of more general central limit theorems for additive functionals of Markov chains, i.e. for expressions of the form $$S_n = \frac 1 {\sqrt n}\sum_{k=1}^n [f(X_k)-\pi(f)]$$ where $(X_n)$ is an ergodic Markov chain with invariant measure $\pi$. Here the chain is the sequence of jumps, and he function is the identity. Possible reference: G. Jones, "On the Markov chain central limit theorem", Probability Surveys 1 (2004), 299-320 (link).

About estimating the diffusion parameter, the increments of your walk are orthogonal in distribution, so you can compute the expected square norm of $S_n$ directly by expanding the sum...

Maybe this can be done by hand, but if not, it is a consequence of more general central limit theorems for additive functionals of Markov chains, i.e. for expressions of the form $$S_n = \frac 1 {\sqrt n}\sum_{k=1}^n [f(X_k)-\pi(f)]$$ where $(X_n)$ is an ergodic Markov chain with invariant measure $\pi$. Here the chain is the sequence of jumps, and he function is the identity. Possible reference: G. Jones, "On the Markov chain central limit theorem", Probability Surveys 1 (2004), 299-320 (link).

About estimating the diffusion parameter, the increments of your walk are orthogonal in distribution, so you can compute the expected square norm of $S_n$ directly by expanding the sum...

Maybe this can be done by hand, but if not, it is a consequence of more general central limit theorems for additive functionals of Markov chains, i.e. for expressions of the form $$S_n = \frac 1 {\sqrt n}\sum_{k=1}^n [f(X_k)-\pi(f)]$$ where $(X_n)$ is an ergodic Markov chain with invariant measure $\pi$. Here the chain is the sequence of jumps, and he function is the identity.

About estimating the diffusion parameter, the increments of your walk are orthogonal in distribution, so you can compute the expected square norm of $S_n$ directly by expanding the sum...

Maybe this can be done by hand, but if not, it is a consequence of more general central limit theorems for additive functionals of Markov chains, i.e. for expressions of the form $$S_n = \frac 1 {\sqrt n}\sum_{k=1}^n [f(X_k)-\pi(f)]$$ where $(X_n)$ is an ergodic Markov chain with invariant measure $\pi$. Here the chain is the sequence of jumps, and he function is the identity. Possible reference: G. Jones, "On the Markov chain central limit theorem", Probability Surveys 1 (2004), 299-320 (link).

About estimating the diffusion parameter, the increments of your walk are orthogonal in distribution, so you can compute the expected square norm of $S_n$ directly by expanding the sum...