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I am looking for a reference for a local CLT for the usual SRW on the cycle -- in continuous-time, ideally. I know the statement for a SRW (and a reference, say Lawler and Limic; Random Walk: A Modern Introduction). As such, with a bit of algebra, I'm sure I can translate this into a statement about the cycle, $\mathbb Z_m$ say. However, a reference would be preferable -- not only as it's less for me to do, but from a publishing point of view.

Informally, I feel that the scaling limit of the SRW on $\mathbb Z_m$ evaluated at time $s = \alpha m^2$, for $\alpha \in \mathbb R$, should be a normal distribution taken modulo 1 (with the correct variance).

If anyone knows of a reference for a precise statement of this result, I would be most appreciative.

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