This thread is funny! But I will take the bait.
US air force is a big fan of the energy maneuverability theory. It requires the fighters to maintain high energy during the dog fight. It has settled down to about 42 degree wings and very powerful engine for sustained turn rate. I doubt J-20 can out-turn F-22 on sustained turn.
However, there are different tactics in dog fight. Even with 3G fighters, Mirage-2000 is a prime example. It has amazing ability to reduce speed to achieve the instant turn rate. This is also reflected on USN's F/A-18. Between the exercises between F-16 and F/A-18, (based on an article I read, on Air Force magazine, by a navy pilot), it is very rare that F-16 can get a first shot against F/A-18. The writer himself never lost a fight against F-16. Usually, F-16's tactic is to accelerate way past, hoping to get out of the range of the short-range missile, then turn.
By the way, wasn't F/A-18 the only 3G fighter which had a kill against F-22 in the exercises?
J-20's delta wing (like Typhoon, Rafael, Mirage-2000) has a very high drag when turning, thus can reduce the speed and turn quickly.
So, if we compare the ability to dog fight, I would say J-20 will get the first shot. F-22 possible would use the same tactic as F-16 vs F/A-18, blaze past then turn after 10 miles or so. If it can evade the first missile, it has the advantage over J-20 on energy and J-20 would have little chance.
By the way, that scenario is one-against-one. If it is a group fight, it would be suicidal to reduce speed to turn. In that case, the J-20's chance is much lower unless it gets super powerful engine (WS-15?). But what's the chance that 4G fighters have a group dog fight?