Donsker's theorem for self-normalized partial sums processes

Let X, X1, X2,... 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 max1≤j≤n |Xj|/Vn → P 0, as n → ∞, where Vn2 = ∑j=1n Xj2.