Atmospheric pressure plasma jets driven by AC voltage are reported to exhibit randomness or periodicity under certain operating conditions. In the present article, discharges in a helium plasma jet driven by 16 kHz sinusoidal voltage are studied. In the positive half cycle, several short and slow periodic discharges (3–6 km/s) are observed before the ignition of random longer and faster bullets (20–60 km/s). The characteristics of the random discharges are studied using statistical methods. The propagation of slow plasma bullets and fast bullets is analysed using an ICCD camera and a streak camera. Glowing like plasma that lasts for 2–6 μs is observed near the gap region and possibly provides a high seed electron density for a following microdischarge. The results are compared with a previous study with a similar APPJ which exhibits multi-periodicity under certain conditions. It is proposed that volumetric residual charges may play an important role for APPJs with short gap distances.