It appears that you'll be demoted to a lower arena if you perform badly. However, I wasn't demoted immediately from arena 3 when I reached lower then 800 trophies, while 800 is the limit to be promoted into arena 3.
What is the limit for each arena to be demoted?
Best Answer
You get officially demoted when you reach 50 trophies below the threshold for a given Arena. You will receive an in-game notification when this occurs. Similarly, you get promoted when you reach you cross the threshold listed for a given Arena. You can view the thresholds for all Arenas at any time by tapping the picture of the Arena you are currently in above the "Battle" button on the Battle tab.
As others have noticed, you can visit the Arena above or below your Arena, if you are close to the threshold and get matched with someone who has qualified for that Arena. Chests you are awarded for victory will be locked to the Arena where the match took place.
Note however, that the Free Chest, Crown Chest, and Shop will all reflect Cards and Gold according to the Arena you are actually qualified for. This is the main difference of being on one side or the other of the threshold.