  We study the effect of drift in pure-jump transaction-level models for asset
prices in continuous time, driven by point processes. The drift is as-sumed to
arise from a nonzero mean in the efficient shock series. It follows that the
drift is proportional to the driving point process itself, i.e. the cumulative
number of transactions. This link reveals a mechanism by which properties of
intertrade durations (such as heavy tails and long memory) can have a strong
impact on properties of average returns, thereby poten-tially making it
extremely difficult to determine long-term growth rates or to reliably detect
an equity premium. We focus on a basic univariate model for log price, coupled
with general assumptions on the point process that are satisfied by several
existing flexible models, allowing for both long mem-ory and heavy tails in
durations. Under our pure-jump model, we obtain the limiting distribution for
the suitably normalized log price. This limiting distribution need not be
Gaussian, and may have either finite variance or infinite variance. We show
that the drift can affect not only the limiting dis-tribution for the
normalized log price, but also the rate in the corresponding normalization.
Therefore, the drift (or equivalently, the properties of dura-tions) affects
the rate of convergence of estimators of the growth rate, and can invalidate
standard hypothesis tests for that growth rate. As a rem-edy to these problems,
we propose a new ratio statistic which behaves more
