In that case, to even start the discussion, you need to define your terms. What exactly do you mean by reincarnated, and what exactly is being reincarnated? Before even pondering evidence, you need to be clear on what you are trying to support.
By reincarnated I mean that there's a connection between the person the current person remembers being and their current self wherein the memories of the memories of the former are had by the latter.
By being reincarnated I mean the memories of the previous person show up in someone else. (I know that this is where they usually say awareness is transferred from one body to another, but let's stick with memories)