It's pretty much guaranteed.
I calculated the number of my direct ancestors back through successive generations: 2 born in the 1930s, 4 born in the 1900s-8 born in the 1870s-16-32, etc.
By the 1100s the number of ancestors is more than the population of Italy at the time. By the 900s it's more than the population of the world.
If any Roman soldier left descendants still alive today we both are statistically certain to have descended from him.
“I calculated the number of my direct ancestors back through successive generations: 2 born in the 1930s, 4 born in the 1900s-8 born in the 1870s-16-32, etc. By the 1100s the number of ancestors is more than the population of Italy at the time. By the 900s it’s more than the population of the world.”
It does not work like that. Your math is wrong. The further you go back, the greater the likelihood of there being an overlap. In order to avoid any overlap would have required everyone to marry only the most distant unrelated person on the planet, which rarely happened, if ever. This is why there are nations, languages, ethnicities, and clans / tribes.
You would have to go back much further before you would find a specific common ancestor, such as a specific Roman soldier. Sure, if you are white, there is a decent chance that somewhere in your lineage was a Roman soldier. That does not mean it would be the same Roman soldier that is in the lineage of another white person.
“If any Roman soldier left descendants still alive today we both are statistically certain to have descended from him.”
Only if a specific Roman soldier has a massive number of descendants and others do not. If the “Smiths” had a specific soldier as an ancestor, and assuming these “Smiths” are all related by a common ancestor since the time of that soldier (rather than the name being common because their profession was common) then it would be possible. But considering all factors, it seems highly unlikely rather than “pretty much guaranteed.”
It is even more unlikely that Asian or the people indigenous to North, South, and Central America were descended from a Roman soldier. Their posterity would not have been either, unless some of their ancestors intermarried with Europeans.
If someone is Italian (as per your example of the population of Italy in the 1100s), that person can be certain to have descended from at least a certain segment of the Italians of the 1100s—possibly a large segment. But this is because that was an isolated population and the person inquiring knows he or she is Italian. As a white American of mostly British ancestry, it is quite possible I also have Italian ancestors from the 1100s. But it is highly unlikely that I have any Asian ancestors from the 1100s.
Wow.
I read once that all European descended people were related at least as closely as 32nd cousins, or something like that.