I am a mathematics graduate student, and I am searching for a (preferably paid) research internship that could last to 6 months in the field of probability theory. I have searched for opportunities in Europe, mainly universities (Oxford, Cambridge, ETH,...) but I am finding it a bit hard to find a long internship (the most popular ones are summer internships for undergraduate students). Also, I did not find any independent research centers that explicitly work in this field.
Therefore, I am asking for help in this matter. I do not know if it is possible to contact professors and ask them to organize internships (especially in the UK) and how I would approach this matter (do I just send an email asking to do an internship, or should I specify my professional project?).