As proved in Donsker-Prohorov's Invariance principle, for i.i.d random variables $\xi_1,\xi_2,\cdots$, its partial sums $S_n(t)=\sum_{i=1}^{[nt]}\xi_i$ converge to Brownian motion $W(t)$ in distribution.
What if the random variables $\xi_1,\xi_2,\cdots$ are independent but not identically distributed(i.e, same mean as 0 but different variance $v_i^2$)? Did someone prove a similar result for not identically distributed cases?
If not, how should I prove this by following the original proof?
I would appreciate if someone can help!
When the sequence is not identically distributed you need to define $S_n$ differently, and then the theorem holds true under suitable conditions for martingales and under some mixing conditions, without the independence.
See Theorem 3 in the following paper
https://projecteuclid.org/journals/annals-of-probability/volume-6/issue-2/Stopping-Times-and-Tightness/10.1214/aop/1176995579.full
Take there $k_n(t)$ to be the first time $k$ the variance of the partial sum $S_k$ is greater or equal than $t$ times the variance of $S_n$.
See also two recent papers about the subject
https://projecteuclid.org/journals/bernoulli/volume-25/issue-4B/Functional-CLT-for-martingale-like-nonstationary-dependent-structures/10.3150/18-BEJ1088.short
and
https://www.sciencedirect.com/science/article/abs/pii/S0167715219302275