The Holy Ghost and Buddha Nature

In some weeks it's Whitsun, one of the less understood Christian days , I think, even by Christians themselves.
It has to do with the Holy Ghost, the third partner of the Trinity Father (God), Son (Jesus) and Holy Ghost.
My question is: when as an ex-Christian I try to understand the Holy Ghost concept;
and when I try as a Theravadin try to understand the (Mahayana) Buddha Nature concept,
I see strong similarities between these two concepts. Is that correct?
Both are already inside a sentient being AND is coming from outside, both are a kind of sparkle of God.
Perhaps the Holy Ghost concept is a bit more reified then the Buddha Nature concept (although I have texts in which Buddha Nature is God-like), both are doing the same function within a person: inspiring.
But perhaps I'm looking (as an agnostic outsider regarding these concepts) to superficial