There is the same person, but you shouldn't see a person as a thing. It's a process. Liken it to a song: the notes fade in and out, but still there is a continuity there that is not a "thing". A song is not a thing, it is just a collection of notes. We call it a song, and give it a name, simply because that is useful on a conventional level. If I say I want to listen to Beat It, well, you know what song I mean.
If I call somebody John, that's useful because you know what person I mean. But in the process "John", nothing is constant. You could call the person a "self", and you could have a point, but that's all on the conventional level. In reality there are no concepts, so self and not self don't really apply. We can only see that there is a construction of what we take ourselves to be. With meditation a lot of people shave off a lot of this, making it smaller. The Buddha wanted us to see we are shaving something which has no core. Like taking layers of an onion. You can keep going and nothing is left. You may be stuck on a certain layer right now, and that's ok, but if you keep following the path, you'll find a way to peel it I'm sure.