Resumen de Donsker's theorem for self-normalized partial sums processes

Miklós Csörgo, Barbara Szyszkowicz, Qiying Wu

• 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→P0, as n→∞, where V2n=∑nj=1X2j.