How can I analyse large packets on an Arduino?
All you need to do is analyse the packet on-the-fly using a state machine.

http://www.gammon.com.au/statemachine

You certainly don't need to hold the entire packet at once, however you do need room for the information the packet contains.

Say, for example, that the packet, although wordy, contains 4 x 32-bit numbers of interest. All you need memory for is those 4 numbers, plus a minor extra amount for the current data you are processing (eg. the current state, and the current number being assembled).

Conceptually it is like streaming a movie to your PC. The PC doesn't need to hold the whole movie, it just needs to hold the current frame, and a few seconds in advance, just to keep things running smoothly.