On the invariance principle for signed-rank statistics

Abstract

continuous a.e. [#], the limit here being in the sense of weak convergence of distributions. Lamperti [5] improved Donsker's result in the following way: he showed that (1) holds for a much larger class of functionals, provided further assumptions are made on the moments of the Xi's. This was achieved by replacing the space C[0, 1] of continuous functions by the space Lip, of functions which are H61der continuous of order c~. If the Xi's have finite moments of all orders, then Corollary 1 of Lamperti [5] says that (1) holds for any functional which is continuous in the topology of Lips, a.e. [/~] on Lip~, for some 7 89 A result analogous to Donsker's invariance principle was proved by Sen [8] for signed-rank statistics. In its general form Sen's theorem is false (Sen [9]). However Sen's proof is correct for the statistics considered here.