Donsker's theorem for self-normalized partial sums processes

Abstract

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 $V_n^2=\sum_{j=1}^{n}X_j^2$.