The abbreviation for "independent, identically distributed" is "IID".
Is there a standard abbreviation for the narrower concept "independent, uniformly distributed"?
"IUD" is the obvious choice, but I don't think this abbreviation is common, so even in a mathematical text, readers might have to momentarily think, "oh, not the birth control method, i.e. intrauterine device". "UID" is another option, but the phrase "uniform, independently distributed" doesn't map as naturally to the broader "independent, identically distributed".
(Seems like this might be a duplicate, but I couldn't find an earlier version.)
I don't think I've ever seen an abbreviation for this in the literature.
If you mean the continuous uniform distribution, you can of course write something like "iid $U(0,1)$", or "independent $U(a_n, b_n)$" if they don't all have the same distribution. Otherwise I would just write "independent uniform".