The Action Cam recorder has an input voltage range of 12-34V, and has a power consumption of 30W during recording and 18W when in standby mode.
The ARRI ALEXA RS power outputs can be used to provide a 24V output to the recorder, and with this input voltage the current drawn by the recorder is 1.2A. If powering other accessories from the cameras RS outputs it is worth noting that they can support a combined load of 2.2A.
The ARRI ALEXA 12V output is not suitable for powering the Action Cam recorder.