Invariance principles for adaptive self-normalized partial sums processes

Let zeta (se)(n) be the adaptive polygonal process of self-normalized partial sums S-k = Sigma (1 less than or equal toi less than or equal tok) X-i of i.i.d. random variables defined by linear interpolation between the points (V-k(2)/V-n(2),S-k/V-n), k less than or equal to n, where V-k(2)=Sigma (i less than or equal tok) X-i(2). We investigate the weak Holder convergence of zeta (se)(n) to the Brownian motion W. We prove particularly that when X-1 is symmetric, zeta (se)(n) converges to W in each Holder space supporting W if and only if X-1 belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lamperti's FCLT where a moment of X-1 of order p > 2 is requested for some Holder weak convergence of the classical partial sums process. We also present some partial extension to the nonsymmetric case. (C) 2001 Elsevier Science B.V. All rights reserved.