  In quantum transport through nanoscale devices, fluctuations arise from
various sources: the discreteness of charge carriers, the statistical
non-equilibrium that is required for device operation, and unavoidable quantum
uncertainty. As experimental techniques have improved over the last decade,
measurements of these fluctuations have become available.} They have been
accompanied by a plethora of theoretical literature using many different
fluctuation statistics to describe the quantum transport. In this paper, we
overview three prominent fluctuation statistics: full counting, waiting time,
and first-passage time statistics. We discuss their weaknesses and strengths,
and explain connections between them in terms of renewal theory. In particular,
we discuss how different information can be encoded in different statistics
when the transport is non-renewal, and how this behavior manifests in the
measured physical quantities of open quantum systems. All theoretical results
are illustrated via a demonstrative transport scenario: a Markovian master
equation for a molecular electronic junction with electron-phonon interactions.
{{} We demonstrate that to obtain non-renewal behavior, and thus to have
temporal correlations between successive electron tunneling events, there must
be a strong coupling between tunneling electrons and out-of-equilibrium
quantized molecular vibrations.
