Convergence of self-normalized partial sums processes in C[0,1] and D[0,1]

Let $(X_i)_{i\geq 1}$ be an i.i.d. sequence of mean zero random variables, $S_n:= X_1+\cdots + X_n$ and $V_n^2:=X_1^2+\cdots +X_n^2$. We consider four sequences of partial sums processes: the broken lines with vertices at the points $(k/n,S_k/V_n)$ or $(V_k^2/V_n^2,S_k/V_n)$ and the corresponding random step functions. We prove that each of them converges weakly in $C[0,1]$ or $D[0,1]$ to the Brownian motion \emph{if and only if} $X_1$ belongs to the \emph{domain of attraction of the normal distribution}. These results contrast with the classical Donsker Prohorov invariance principles where the N.S.C. for such convergences is $\E X_1^2 < \infty$.