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Recent results for getting diffusion limits of a sequence of suitably scaled stochastic processes are applied to the synchronization problem for a digital phase locked loop (DPLL). For small filter gains and symbol intervals a diffusion (Gauss-Markov) process approximation is rigorously obtained. This approximation yields approximate error variances, passage time distributions, and correlation properties (among other properties) for the DPLL. The problem of tracking a drifting clock is also treated.