Correction, it keeps a rolling loop going so that when the CPU hears the wake word it has enough audio buffered for voice recognition to have a clean starting point.

If it only started recording when the wake word was processed as being heard there would be an added delay before you could start speaking a command. Might as well push a button and wait for a beep for that added delay.

And the wake word detection is done offline, as can be easily proven by disconnecting your internet and saying 'Alexa report this conversation to the NSA' - she will wake up, but then blink red due to lack of network connectivity.