As far as bow and arrows go, historians don't favor a diffusionist approach. The evidence we have is old enough, and the materials for a bow perishable enough, that it's a very challenging thing to know with any certainty. Bows appear far too late in cenral North America, though, for them to have reasonably come across from Asia.

Swords, spears, and the like, however, are very simple (and, if we include blunt maces, the simplest) ways to solve the same problem. It's no suprise that they would pop up in various places. You read more about this from an engineering perspective and an anthropology one.