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In this paper we show that the continuous version of the self normalisedprocess $Y_{n,p}(t)= S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p}$ 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 belongto $DA(\alpha)$, has a non trivial distribution iff $p=\alpha=2$. The case for$2 > p > \alpha$ and $p \le \alpha < 2$ is systematically eliminated by showingthat either of tightness or finite dimensional convergence to a non-degeneratelimiting distribution does not hold. This work is an extension of the work byCs\"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 standardBrownian motion in sup norm.

Process convergence of self normalized sums of i.i.d. random variables coming from domain of attraction of stable distributions