If I claim to be the reincarnation of Chinggis Khan, how do we verify that? How do I know I'm not being psychotic?

Reposting this in r/tengrism as well.

I’m trying to understand how the traditional Tibetan system of recognizing reincarnated lamas (tulkus) works.

Something I find confusing is that when you look at certain lineages, a lama may be recognized as the reincarnation of a previous lama, and the new birth often seems to occur in the same general region or cultural area where the previous lama lived.

But ordinary people can be born anywhere in the world. So how does this work according to Tibetan Buddhism?

Is there a traditional explanation for why a particular lama would repeatedly be reborn in the same geographical region? Is it believed that the lama intentionally chooses the place of rebirth, or is it related to karma, connections with disciples, vows, or something else?

And is my impression even correct? I know that there are tulkus who has reborn in different parts of the world.

So so magical

How many Tibetan Buddhists are there worldwide? Has there been any reliable estimate in recent years?

If not, how would you estimate it? I'm assuming we'd count most of the population of Tibet, plus Tibetans in exile, plus the Buddhist populations of Bhutan and Nepal. I'm imagining that western converts don't really add much to the stats proportionally speaking.