Invariance principles for adaptive self-normalized partial sums processes

Let zeta (se)(n) be the adaptive polygonal process of self-normalized parti al sums S-k = Sigma (1 less than or equal toi less than or equal tok) X-i o f i.i.d. random variables defined by linear interpolation between the point s (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 supporti ng W if and only if X-1 belongs to the domain of attraction of the normal d istribution. 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 cl assical partial sums process. We also present some partial extension to the nonsymmetric case. (C) 2001 Elsevier Science B.V. All rights reserved.