Donsker's Invariance for independent, identical mean, variance random variables?

Question

Is there an extension of standard Donsker's Invariance Principle to a sequence of independent random variables with zero mean and unit variance but not necessarily identically distributed?

Answer 1

Repeat? Is there an extension of Donsker's invariance principle for not identically distributed random variables?

You want convergence to hold on $C[0,1]$, so you might look into general CLTs. They have different ways of going about writing sufficient conditions, but it always comes down to a) something called tightness, which is about concentrating on compact sets and b) convergence at each point in $[0,1]$. The former will follow if your scaled partial sums are uniformly bounded in probability, and the latter will be easy from an i.n.i.d. CLT like Lindeberg-Feller.

Triangular arrays more than take care of non-identical RVs.

Here are some general CLTs

Andersen, Gine, Zinn (allows for infinite variance also): https://www.ams.org/journals/tran/1988-308-02/S0002-9947-1988-0930076-3/S0002-9947-1988-0930076-3.pdf

Araujo, Gine: https://books.google.com/books/about/The_Central_Limit_Theorem_for_Real_and_B.html?id=3exhnQAACAAJ