Invariance principles for adaptive self-normalized partial sums processes

Let ζnse be the adaptive polygonal process of self-normalized partial sums Sk=∑1⩽i⩽kXi of i.i.d. random variables defined by linear interpolation between the points (Vk2/Vn2,Sk/Vn), k⩽n, where Vk2=∑i⩽k Xi2. We investigate the weak Hölder convergence of ζnse to the Brownian motion W. We prove particularly that when X1 is symmetric, ζnse 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 Lampertiʹs 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.