  In this paper we show that the continuous version of the self normalised
process $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ where
$S_n(t)=\sum_{i=1}^{[nt]} X_i$ and $V_{(n,p)}=
\sum_{i=1}^{n}|X_i|^p)^{\frac{1}{p}}$ and $X_i$ i.i.d. random variables belong
to $DA(\alpha)$, has a non trivial distribution iff $p=\alpha=2$. The case for
$2 > p > \alpha$ and $p \le \alpha < 2$ is systematically eliminated by showing
that either of tightness or finite dimensional convergence to a non-degenerate
limiting distribution does not hold. This work is an extension of the work by
Cs\"org\"o et al. who showed Donsker's theorem for $Y_{n,2}(\cdot)$, i.e., for
$p=2$, holds iff $\alpha =2$ and identified the limiting process as standard
Brownian motion in sup norm.
