Ireland, Society Of Friends (Quaker) Family Lists, image, FindMyPast (https://search.findmypast.co.uk/record/browse?id=ire%2fquaker%2fmmv111m-1%2f0027 : accessed 24 October 2019), marriage of Samuel Randall and Rebeckah Atkins in Cork on 5d 9mo (Nov) 1683; citing Cork MM Family lists 1671-1872, Religious Society Of Friends In Ireland Archives.
Ireland, Society Of Friends (Quaker) Family Lists, image, FindMyPast (https://search.findmypast.co.uk/record/browse?id=ire%2fquaker%2fmmv111m-1%2f0027 : accessed 24 October 2019), death of Rebeckah Randall on 13d 1mo (Mar) 1739 in George Pease's, Limerick; citing Cork MM Family lists 1671-1872, Religious Society Of Friends In Ireland Archives.
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DNA Connections
It may be possible to confirm family relationships with Rebecca by comparing test results with other carriers of her mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Rebecca:
Atkins-5589 and Atkinson-13499 appear to represent the same person because: Had a different last name in Family Search. Atkins seems reasonable. No additional info in (erroneously) newer profile.
Atkins-5589 and Atkinson-13499 do not represent the same person because: Sorry, but these are different people. My Rebecca Atkins was born in Ireland in 1661 -- I have her original birth and death record. Your Rebecca Atkinson was born 8 years earlier and has a different last name.
Further to my last message, I went back to the original Quaker records and noticed that one of the transcriptions of the marriage had her name as Atkinson rather than Atkins. This is probably where the erroneous Familysearch record originated. However, this was clearly an error because the original marriage record clearly states that her name was Rebecca Atkins. Therefore, I don't think Rebecca Atkinson existed. If you like, I can merge the profiles.