It's quite possible that the bow and arrow was invented once, and the idea spread from that one source.

It isn't easy to trace the evolution of the bow archaeologically, since wood does not survive well over many millennia. Projectile points can provide clues as to whether the bow and arrow were present, but while small points might be arrow heads, they could also be javelin heads. Also, wood and bamboo arrowheads are common, so an absence of stone arrow points in the archaeological record doesn't mean that the bow and arrow were not in use.

Further, the bow and arrow might be known by a people, but not used significantly. The spearthrower and the sling (and blowpipe) are competing technologies that are sometimes the choice for hunting and warfare, while the bow is used, e.g., as a children's toy. The appearance of the bow and arrow in the archaeological (or historical) record in an area where it wasn't in use, nor in use by neighbours, doesn't necessarily mean a new independent invention.

Based on stone/bone points that might be projectile points and arrow heads in particular, archery might be over 60,000 years old. The oldest bows that have been found are about 10,000 years old (and fragments thought to be from bows have been found, about 12,000 years old). If the older dates are correct, it's quite possible that there was a single invention.

A past answer of mine on the invention of the bow and its diffusion:

• https://www.reddit.com/r/AskHistorians/comments/9pgvu3/where_did_the_bow_and_arrow_originate_and_how_did/

For the spread of archery in the Americas, see the answer by u/retarredroof in:

• https://www.reddit.com/r/AskHistorians/comments/3l1iif/did_native_americans_use_the_bow_and_arrow_prior/

Note that the projectile point evidence suggests that the bow and arrow disappeared and reappeared in some areas. it is possible that these reappearances were due to independent reinventions, but it is also quite possible that the bow and arrow did not completely disappear between periods of common use.

There are particular types of bows that were developed independently in different parts of the world. For example, the sinew-horn-wood composite bow was invented in both Eurasia and North America, and paddle bows were invented in both Asia and America.