Invariance principles for adaptive self-normalized partial sums processes

Let [zeta]nse be the adaptive polygonal process of self-normalized partial sums Sk=[summation operator]1[less-than-or-equals, slant]i[less-than-or-equals, slant]kXi of i.i.d. random variables defined by linear interpolation between the points (Vk2/Vn2,Sk/Vn), k[less-than-or-equals, slant]n, where Vk2=[summation operator]i[less-than-or-equals, slant]k Xi2. We investigate the weak Hölder convergence of [zeta]nse to the Brownian motion W. We prove particularly that when X1 is symmetric, [zeta]nse converges to W in each Hölder space supporting W if and only if X1 belongs to the domain of attraction of the normal distribution. This contrasts strongly with Lamperti's FCLT where a moment of X1 of order p>2 is requested for some Hölder weak convergence of the classical partial sums process. We also present some partial extension to the nonsymmetric case.