  We theoretically analyse an autonomous clock comprising a
nanoelectromechanical system, which undergoes self-oscillations driven by
electron tunnelling. The periodic mechanical motion behaves as the clockwork,
similar to the swinging of a pendulum, while induced oscillations in the
electrical current can be used to read out the ticks. We simulate the dynamics
of the system in the quasi-adiabatic limit of slow mechanical motion, allowing
us to infer statistical properties of the clock's ticks from the current
auto-correlation function. The distribution of individual ticks exhibits a
tradeoff between accuracy, resolution, and dissipation, as expected from
previous literature. Going beyond the distribution of individual ticks, we
investigate how clock accuracy varies over different integration times by
computing the Allan variance. We observe non-monotonic features in the Allan
variance as a function of time and applied voltage, which can be explained by
the presence of temporal correlations between ticks. These correlations are
shown to yield a precision advantage for timekeeping over the timescales that
the correlations persist. Our results illustrate the non-trivial features of
the tick series produced by nanoscale clocks, and pave the way for experimental
investigation of clock thermodynamics using nanoelectromechanical systems.
