Let $X, X_1, X_2,\ldots$ be a sequence of nondegenerate i.i.d. random
variables with zero means. In this paper we show that a self-normalized
version of Donsker's theorem holds only under the assumption that X
belongs to the domain of attraction of the normal law.
A thus resulting extension of the arc sine law is also discussed.
We also establish that a weak invariance principle holds true for
self-normalized, self-randomized partial sums processes of independent
random variables that are assumed to be symmetric around mean zero,
if and only if $\max_{1\le j\le n}|X_j|/V_n\to_P 0$, as $n\to\infty$,
where $V_n^2=\sum_{j=1}^{n}X_j^2$.