hi! This question may be worth x-posting to /r/AskAnthropology. FYI, there have been many posts asking about the development of bow & arrow technology; you may be interested in the previous discussions

• Who invented the bow and arrow?

• Was the bow and arrow invented once, or many times across human history?

• What society first used the bow-and-arrow? And did the technology spread or was it largely discovered and developed independently across the globe? Lastly, were there any major civilizations that did not employ the bow?

• Did the bow and arrow develop independently in the various cultures across the globe? If so, why does the design seem so universally similar?

• Did the bow and arrow migrate around the world with moving peoples or was it invented independently by hundreds of different societies?

• How come every civilization - even the secluded Sentinelese - discovered/invented bows and arrows?

• Did the bow and arrow develop before the land bridge between Alaska and Russia disappeared or did people in Eurasia and the Americas create the bow independent of each other?

• How did Native Americans develop bow-and-arrow technology?

• Did the Native Americans have bow & arrows by 1000BC?

• Is it just a coincidence that both native americans and europeans had bows before the columbian exchange?

• When were the first bow and arrows used, and how did many cultures develop them without ever coming into contact with each other?

• Why was the bow and arrow design used throughout the world even when they had little contact with each other?

• Why did the Australian aborigines never invent bows and arrows? Why did nobody else invent the boomerang?

It is believed that the bow and arrow were developed independently at a number of locations about 3-4 thousand years ago. See this post. So, no paleoindians did not come to the Americas equipped with the bow and arrow.

Edit: after further reading, bow and arrow probably developed 2500-3000 years ago in the northern portion of North America and 1500-100 years BP in western North America. See this article for further discussion.