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

In this paper we show that the continuous version of the self-normalized process Y-n,Y-p(t) = S-n(t)/V-n,V-p + (nt- [nt1)X-[nt]+1/V-n,V-p, 0 < t <= 1; p > 0 where S-n(t) =Sigma([nt]) X-i=1(i) and V-(n,V-p) =(Sigma(n)(i=1) vertical bar Xi(vertical bar) (P))(1/P) and X-i i.i.d. random variables belong to Dit(a), has a non-trivial distribution iff p = alpha = 2. The case for 2> p > alpha and p <= 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 Csorgo et al. who showed Donsker's theorem for Y-n,Y-2 (.) i.e., for p = 2, holds if alpha = 2 and identified the limiting process as a standard Brownian motion in sup norm.