Can previous villagers from Animal Crossing: New Leaf move back in?

My favourite character from Animal Crossing: New Leaf, Chief, moved out about a month and a half ago. Some people are saying old villagers who moved out can move back in again. Can previous villagers move back in to my town? And if so, can I do anything to make it happen?

This FAQ on Reddit specifically around New Leaf says:

After you’ve had 4 villagers move out, the 16 most recent villagers to have left your town have the chance to show up on main street to do a little shopping every day. They just miss you! If you don’t want them there, get 16 more villagers to move out and the one you don’t like will be gone. This is also a good indicator of whether an old villager can move back into your town or not. The moment they stop appearing on main street, they’ll have cleared the 16 villager cycle and will be able to move back in to your town.

Now about moving in, I do know that you cannot convince then to move back even if you see them shopping about. You'll have to get them by random chance again or find the desired villager in a friend's town who is moving out.