The name of invariance principle of Donsker

Question

I have seen the invariance principle of Donsker for the Wiener measure in Karatzas' Brownian Motion and Stochastic Calculus. I am wondering why this theorem have this name, e.g. where does the invariance come from? Thanks!

Answer 1

"Invariance" here means that the result does not depend on the law of $X_1$ provided that $X_1$ is centered and has a finite moment. That is, the convergence $$\left(n^{-1/2} \sum_{j=1}^{[nt]}X_j\right)_{t\in [0,1]} \to (W_t)_{t\in[0,1]} $$ holds if $X_1$ is centered and has a finite moment. We get a Wiener process as a limit even if the law of $X_1$ is far from being Gaussian. In particular, the moments of order $2+\delta$ may fail to exists for any positive $\delta$.