No matter where you look, the bow is a weapon that can be found in the hands of many soldiers from around the world. Even uncontacted tribes, completely cut off from the outside world, have been seen holding bows.
While I don't think it's possible to rule out that the bow may have been independently invented more than once, its invention long predates civilization. Bows have been used since the upper paleolithic. People took the prehistoric knowledge of bows and arrows with them when they migrated around the world before civilization arose.
The Australian Aborigines are a notable exception to humans having prehistoric knowledge of archery, having migrated very early (50,000 BC or thereabouts) and been isolated from more mainstream trends until recently.
FYI, you'll find more info on this subject here
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