It was my understanding that the projection comes simply from extrapolating the figures from the past few decades, apart form looking at the age pyramid of Europe and seeing how much immigration will be needed to pay for pensions, health care and so on. I haven't done the math myself so I can't tell you whether it is true. If this is a collaborative effort, then I think it would be wise for someone with the background to dismiss those projections mathematically.

However to show that such a demographic change is not such a statistical impossibility, look at US demographics. Ethnic Europeans were around 80% of the US population in the 80s, they are around 60% now and it is projected by the US government itself that in the next few decades they will be below 50%. You can bump that last percentage a little bit to account for those counted as Hispanic whites who are actually predominantly indigenous Europeans. I know US demographics are completely unrelated and have nothing to do with the OP which relates to Europe. This was simply to show how such demographic changes are not impossible. Nevertheless, there could be other variables which I'm not aware of, that make what we've seen in the US possible but impossible in Europe.

For that matter, birthrates are plummeting in many developing countries including the middle eastern ones - this is generally not acknowledged in these kind of projections.