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 assumed to arise from a non‐zero mean in the efficient shock series. It follows that the drift is proportional to the driving point process itself, that is, 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 potentially 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 memory 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 distribution for the normalized log price but also the rate in the corresponding normalization. Therefore, the drift (or equivalently, the properties of durations) affects the rate of convergence of estimators of the growth rate and can invalidate standard hypothesis tests for that growth rate. As a remedy to these problems, we propose a new ratio statistic that behaves more robustly and employ subsampling methods to carry out inference for the growth rate. Our analysis also sheds some new light on two long‐standing debates as to whether stock returns have long memory or infinite variance.