The first known ancestor of our Elg famiy was a Finnish settler who arrived in what is now the village of Säfsnäs in the early 17th Century - Pål Larsson Turpoinen(see http://web.telia.com/~u85435856/history/finnish.html ). Many Swedes with roots in the area can count Pål among their ancestors.
For some years, genealogists have suspected that Pål is identical to a man who grew up in a small village called Pihtipudas in central Finland, and who disappears from Finnish records in 1603.
One clue has been that the name - spelled Turpeinen - still lives on in Pihtipudas. Now, DNA testing has shown that a Swedish descendant of Pål Larsson Turpeinen shares genes with a member of the Pihtipudas Turpeinens, going back to around 1600. So it appears that the origins of Pål Larsson Turpoinen/Turpeinen have been confirmed. And since he disappears from Pihtipudas in 1603, his birth if probably closer to 1580.
When Pål and other Finnish-born settlers arrived in the area they were neither "illegal" nor "immigrants". Finland was part of Sweden since the 13th century, and would remain so for another two centuries, until we lost Finland in a war against Russia in 1809. And the settlers had been invited by the Swedish king, to develop unsettled areas.
You can find the location of Pihtipudas here: https://www.google.com/maps/place/Pihtipudas,+Finland/